From 777a074744579e6f42f5824c7136ca6e1838dcef Mon Sep 17 00:00:00 2001 From: Axel Kohlmeyer Date: Mon, 26 Sep 2022 15:04:57 -0400 Subject: [PATCH 1/4] Convert PDLAMMPS user doc from latex to sphinx --- doc/src/Howto.rst | 1 + doc/src/Howto_peri.rst | 1055 +++++++++++++++++++++++++++++++ doc/src/JPG/dump.peri.2000.png | Bin 0 -> 205139 bytes doc/src/JPG/dump.peri.300.png | Bin 0 -> 147512 bytes doc/src/JPG/dump.peri.600.png | Bin 0 -> 160581 bytes doc/src/JPG/ovito-peri-snap.png | Bin 0 -> 506151 bytes doc/src/JPG/pdlammps_fig1.png | Bin 0 -> 163775 bytes doc/src/JPG/pdlammps_fig2.png | Bin 0 -> 2622167 bytes 8 files changed, 1056 insertions(+) create mode 100644 doc/src/Howto_peri.rst create mode 100644 doc/src/JPG/dump.peri.2000.png create mode 100644 doc/src/JPG/dump.peri.300.png create mode 100644 doc/src/JPG/dump.peri.600.png create mode 100644 doc/src/JPG/ovito-peri-snap.png create mode 100644 doc/src/JPG/pdlammps_fig1.png create mode 100644 doc/src/JPG/pdlammps_fig2.png diff --git a/doc/src/Howto.rst b/doc/src/Howto.rst index 715f13f09b..d1bb1733fc 100644 --- a/doc/src/Howto.rst +++ b/doc/src/Howto.rst @@ -85,6 +85,7 @@ Packages howto Howto_coreshell Howto_drude Howto_drude2 + Howto_peri Howto_manifold Howto_spins diff --git a/doc/src/Howto_peri.rst b/doc/src/Howto_peri.rst new file mode 100644 index 0000000000..6285036fc8 --- /dev/null +++ b/doc/src/Howto_peri.rst @@ -0,0 +1,1055 @@ +Peridynamics with LAMMPS +======================== + +This Howto is based on the Sandia report 2010-5549 by Michael L. Parks, +Pablo Seleson, Steven J. Plimpton, Richard B. Lehoucq, and +Stewart A. Silling. + +Overview +"""""""" + +Peridynamics is a nonlocal extension of classical continuum mechanics. +The discrete peridynamic model has the same computational structure as a +molecular dynamics model. This Howto provides a brief overview of the +peridynamic model of a continuum, then discusses how the peridynamic +model is discretized within LAMMPS as described in the original article +:ref:`(Parks) `. An example problem with comments is also +included. + +Quickstart +"""""""""" + +The peridynamics styles are included in the optional :ref:`PERI package +`. If your LAMMPS executable does not already include the +PERI package, you can see the :doc:`build instructions for packages +` for how to enable the package when compiling a custom +version of LAMMPS from source. + +Here is a minimal example for setting up a peridynamics simulation. + +.. code-block:: LAMMPS + + units si + boundary s s s + lattice sc 0.0005 + atom_style peri + atom_modify map array + neighbor 0.0010 bin + region target cylinder y 0.0 0.0 0.0050 -0.0050 0.0 units box + create_box 1 target + create_atoms 1 region target + + pair_style peri/pmb + pair_coeff * * 1.6863e22 0.0015001 0.0005 0.25 + set group all density 2200 + set group all volume 1.25e-10 + velocity all set 0.0 0.0 0.0 sum no units box + fix 1 all nve + compute 1 all damage/atom + timestep 1.0e-7 + +Some notes on this input example: + +- peridynamics simulations typically use SI :doc:`units ` +- particles must be created on a :doc:`simple cubic lattice ` +- using the :doc:`atom style peri ` is required +- an :doc:`atom map ` is required for indexing particles +- The :doc:`skin distance ` used when computing neighborlists + should be defined appropriately for your choice of simulation + parameters. The *skin* should be set to a value such that the + peridynamic horizon plus the skin distance is larger than the maximum + possible distance between two bonded particles (before their bond + breaks). Here it is set to 0.001 meters. +- a :doc:`peridynamics pair style ` is required. Available + choices are currently: *peri/eps*, *peri/lps*, *peri/pmb*, and + *peri/ves*. The model parameters are set with a :doc:`pair_coeff + ` command. +- the mass density and volume fraction for each particle must be + defined. This is done with the two :doc:`set ` commands for + *density* and *volume*. For a simple cubic lattice, the volume of a + particle should be equal to the cube of the lattice constant, here + :math:`V_i = \Delta x ^3`. +- with the :doc:`velocity ` command all particles are initially at rest +- a plain :doc:`velocity-Verlet time integrator ` is used, + which is algebraically equivalent to a centered difference in time, + but numerically more stable +- you can compute the damage at the location of each particle with + :doc:`compute damage/atom ` +- finally, the timestep is set to 0.1 microseconds with the + :doc:`timestep ` command. + + +Peridynamic Model of a Continuum +"""""""""""""""""""""""""""""""" + +The following is not a complete overview of peridynamics, but a +discussion of only those details specific to the model we have +implemented within LAMMPS. For more on the peridynamic theory, the +reader is referred to :ref:`(Silling 2007) `. To begin, +we define the notation we will use. + +Basic Notation +^^^^^^^^^^^^^^ + +Within the peridynamic literature, the following notational conventions +are generally used. The position of a given point in the reference +configuration is :math:`\textbf{x}`. Let +:math:`\mathbf{u}(\mathbf{x},t)` and :math:`\mathbf{y}(\mathbf{x},t)` +denote the displacement and position, respectively, of the point +:math:`\mathbf{x}` at time :math:`t`. Define the relative position and +displacement vectors of two bonded points :math:`\textbf{x}` and +:math:`\textbf{x}^\prime` as :math:`\mathbf{\xi} = \textbf{x}^\prime - +\textbf{x}` and :math:`\mathbf{\eta} = \textbf{u}(\textbf{x}^\prime,t) - +\textbf{u}(\textbf{x},t)`, respectively. We note here that +:math:`\mathbf{\eta}` is time-dependent, and that :math:`\mathbf{\xi}` +is not. It follows that the relative position of the two bonded points +in the current configuration can be written as :math:`\boldsymbol{\xi} + +\boldsymbol{\eta} = +\mathbf{y}(\mathbf{x}^{\prime},t)-\mathbf{y}(\mathbf{x},t)`. + +Peridynamic models are frequently written using *states*, which we +briefly describe here. For the purposes of our discussion, all states +are operators that act on vectors in :math:`\mathbb{R}^3`. For a more +complete discussion of states, see :ref:`(Silling 2007) +`. A *vector state* is an operator whose image is a +vector, and may be viewed as a generalization of a second-rank +tensor. Similarly, a *scalar state* is an operator whose image is a +scalar. Of particular interest is the vector force state +:math:`\underline{\mathbf{T}}\left[ \mathbf{x},t \right]\left< +\mathbf{x}^{\prime}-\mathbf{x} \right>`, which is a mapping, having +units of force per volume squared, of the vector +:math:`\mathbf{x}^{\prime}-\mathbf{x}` to the force vector state +field. The vector state operator :math:`\underline{\mathbf{T}}` may +itself be a function of :math:`\mathbf{x}` and :math:`t`. The +constitutive model is completely contained within +:math:`\underline{\mathbf{T}}`. + +In the peridynamic theory, the deformation at a point depends +collectively on all points interacting with that point. Using the +notation of :ref:`(Silling 2007) `, we write the +peridynamic equation of motion as + +.. math:: + + + \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x},t) = + \int_{\mathcal{H}_{\mathbf{x}}} \left\{ \underline{\mathbf{T}}\left[ + \mathbf{x},t \right]\left<\mathbf{x}^{\prime}-\mathbf{x} \right> + - \underline{\mathbf{T}}\left[\mathbf{x}^{\prime},t \right]\left<\mathbf{x}-\mathbf{x}^{\prime} \right> \right\} + {d}V_{\mathbf{x}^\prime} + \mathbf{b}(\mathbf{x},t), + +where :math:`\rho` represents the mass density, +:math:`\underline{\mathbf{T}}` the force vector state, and +:math:`\mathbf{b}` an external body force density. A point +:math:`\mathbf{x}` interacts with all the points +:math:`\mathbf{x}^{\prime}` within the neighborhood +:math:`\mathcal{H}_{\mathbf{x}}`, assumed to be a spherical region of +radius :math:`\delta>0` centered at :math:`\mathbf{x}`. :math:`\delta` +is called the *horizon*, and is analogous to the cutoff radius used in +molecular dynamics. Conditions on :math:`\underline{\mathbf{T}}` for +which \eqref{eqn:NewtonII} satisfies the balance of linear and angular +momentum are given in :ref:`(Silling 2007) `. + +We consider only force vector states that can be written as + +.. math:: + + \underline{\mathbf{T}} = \underline{t}\,\underline{\mathbf{M}}, + +with :math:`\underline{t}` a *scalar force state* and +:math:`\underline{\mathbf{M}}` the *deformed direction vector +state*, defined by + +.. math:: + + \underline{\mathbf{M}}\left< \boldsymbol{\xi} \right> = + \left\{ \begin{array}{cl} + \frac{\boldsymbol{\xi} + \boldsymbol{\eta}}{ + \left\Vert \boldsymbol{\xi} + \boldsymbol{\eta} \right\Vert + } & \left\Vert \boldsymbol{\xi} + \boldsymbol{\eta} \right\Vert \neq 0 \\ + \boldsymbol{0} & \textrm{otherwise} + \end{array} + \right. . + +Such force states correspond to so-called *ordinary* materials +:ref:`(Silling 2007) `. These are the materials for which +the force between any two interacting points :math:`\textbf{x}` and +:math:`\textbf{x}^\prime` acts along the line between the points. + + +Linear Peridynamic Solid (LPS) Model +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +We summarize the linear peridynamic solid (LPS) material model. For more +on this model, the reader is referred to :ref:`(Silling 2007) +`. This model is a nonlocal analogue to a classical +linear elastic isotropic material. The elastic properties of a a +classical linear elastic isotropic material are determined by (for +example) the bulk and shear moduli. For the LPS model, the elastic +properties are analogously determined by the bulk and shear moduli, +along with the horizon :math:`\delta`. + +The LPS model has a force scalar state + +.. math:: + + \underline{t} = \frac{3K\theta}{m}\underline{\omega}\,\underline{x} + + \alpha \underline{\omega}\,\underline{e}^{\rm d}, + +with :math:`K` the bulk modulus and :math:`\alpha` related to the shear +modulus :math:`G` as + +.. math:: + + \alpha = \frac{15 G}{m}. + +The remaining components of the model are described as follows. Define +the reference position scalar state :math:`\underline{x}` so that +:math:`\underline{x}\left<\boldsymbol{\xi} \right> = \left\Vert +\boldsymbol{\xi} \right\Vert`. Then, the weighted volume :math:`m` is +defined as + +.. math:: + + m\left[ \mathbf{x} \right] = \int_{\mathcal{H}_\mathbf{x}} + \underline{\omega} \left<\boldsymbol{\xi}\right> + \underline{x}\left<\boldsymbol{\xi} \right> + \underline{x}\left<\boldsymbol{\xi} \right>{d}V_{\boldsymbol{\xi} }. + +Let + +.. math:: + + \underline{e}\left[ \mathbf{x},t \right] \left<\boldsymbol{\xi} + \right> = \left\Vert \boldsymbol{\xi} + \boldsymbol{\eta} + \right\Vert - \left\Vert \boldsymbol{\xi} \right\Vert + +be the extension scalar state, and + +.. math:: + + \theta\left[ \mathbf{x}, t \right] = \frac{3}{m\left[ \mathbf{x} + \right]}\int_{\mathcal{H}_\mathbf{x}} \underline{\omega} + \left<\boldsymbol{\xi}\right> \underline{x}\left<\boldsymbol{\xi} + \right> \underline{e}\left[ \mathbf{x},t + \right]\left<\boldsymbol{\xi} \right>{d}V_{\boldsymbol{\xi}} + +be the dilatation. The isotropic and deviatoric parts of the extension +scalar state are defined, respectively, as + +.. math:: + + \underline{e}^{\rm i}=\frac{\theta \underline{x}}{3}, \qquad + \underline{e}^{\rm d} = \underline{e}- \underline{e}^{\rm i}, + + +where the arguments of the state functions and the vectors on which they +operate are omitted for simplicity. We note that the LPS model is linear +in the dilatation :math:`\theta`, and in the deviatoric part of the +extension :math:`\underline{e}^{\rm d}`. + +.. note:: + + The weighted volume :math:`m` is time-independent, and does + not change as bonds break. It is computed with respect to the bond + family defined at the reference (initial) configuration. + +The nonnegative scalar state :math:`\underline{\omega}` is an *influence +function* :ref:`(Silling 2007) `. For more on influence +functions, see :ref:`(Seleson 2010) `. If an influence +function :math:`\underline{\omega}` depends only upon the scalar +:math:`\left\Vert \boldsymbol{\xi} \right\Vert`, (i.e., +:math:`\underline{\omega}\left<\boldsymbol{\xi}\right> = +\underline{\omega}\left<\left\Vert \boldsymbol{\xi} \right\Vert\right>`\ +), then :math:`\underline{\omega}` is a spherical influence function. +For a spherical influence function, the LPS model is isotropic +:ref:`(Silling 2007) `. + +.. note:: + + In the LAMMPS implementation of the LPS model, the influence function + :math:`\underline{\omega}\left<\left\Vert \boldsymbol{\xi} + \right\Vert\right> = 1 / \left\Vert \boldsymbol{\xi} \right\Vert` is + used. However, the user can define their own influence function by + altering the method "influence_function" in the file + ``pair_peri_lps.cpp``. The LAMMPS peridynamics code permits both + spherical and non-spherical influence functions (e.g., isotropic and + non-isotropic materials). + + +Prototype Microelastic Brittle (PMB) Model +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +We summarize the prototype microelastic brittle (PMB) material +model. For more on this model, the reader is referred to +:ref:`(Silling 2000) ` and :ref:`(Silling 2005) +`. This model is a special case of the LPS model; see +:ref:`(Seleson 2010) ` for the derivation. The elastic +properties of the PMB model are determined by the bulk modulus :math:`K` +and the horizon :math:`\delta`. + +The PMB model is expressed using the scalar force state field + +.. math:: + + \underline{t}\left[ \mathbf{x},t \right]\left< \boldsymbol{\xi} \right> = \frac{1}{2} f\left( \boldsymbol{\eta} ,\boldsymbol{\xi} \right), + +with :math:`f` a scalar-valued function. We assume that :math:`f` takes +the form + +.. math:: + f = c s, + +where + +.. math:: + c = \frac{18K}{\pi \delta^4}, + +with :math:`K` the bulk modulus and :math:`\delta` the horizon, and +:math:`s` the bond stretch, defined as + +.. math:: + + s(t,\mathbf{\eta},\mathbf{\xi}) = \frac{ \left\Vert {\mathbf{\eta}+\mathbf{\xi}} \right\Vert - \left\Vert {\mathbf{\xi}} \right\Vert }{\left\Vert {\mathbf{\xi}} \right\Vert}. + +Bond stretch is a unitless quantity, and identical to a one-dimensional +definition of strain. As such, we see that a bond at its equilibrium +length has stretch :math:`s=0`, and a bond at twice its equilibrium +length has stretch :math:`s=1`. The constant :math:`c` given above is +appropriate for 3D models only. For more on the origins of the constant +:math:`c`, see :ref:`(Silling 2005) `. For the derivation +of :math:`c` for 1D and 2D models, see :ref:`(Emmrich) `. + +Given \eqref{eqn:PMBState}, \eqref{eqn:NewtonII} reduces to + +.. math:: + + \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x},t) = \int_{\mathcal{H}_\mathbf{x}} \mathbf{f} \left( \boldsymbol{\eta},\boldsymbol{\xi} \right){d}V_{\boldsymbol{\xi}} + \mathbf{b}(\mathbf{x},t), + +with + +.. math:: + + \mathbf{f} \left( \boldsymbol{\eta}, \boldsymbol{\xi}\right) =f \left( \boldsymbol{\eta}, \boldsymbol{\xi}\right) \frac{\boldsymbol{\xi}+ \boldsymbol{\eta}}{ \left\Vert {\boldsymbol{\xi} + \boldsymbol{\eta}} \right\Vert}. + +Unlike the LPS model, the PMB model has a Poisson ratio of +:math:`\nu=1/4` in 3D, and :math:`\nu=1/3` in 2D. This is reflected in +the input for the PMB model, which requires only the bulk modulus of the +material, whereas the LPS model requires both the bulk and shear moduli. + +.. _peridamage: + +Damage +^^^^^^ + +Bonds are made to break when they are stretched beyond a given +limit. Once a bond fails, it is failed forever :ref:`(Silling) +`. Further, new bonds are never created during the course +of a simulation. We discuss only one criterion for bond breaking, called +the *critical stretch* criterion. + +Define :math:`\mu` to be the history-dependent scalar +boolean function + +.. math:: + + \mu(t,\mathbf{\eta},\mathbf{\xi}) = \left\{ + \begin{array}{cl} + 1 & \mbox{if $s(t^\prime,\mathbf{\eta},\mathbf{\xi})< \min \left(s_0(t^\prime,\mathbf{\eta},\mathbf{\xi}) , s_0(t^\prime,\mathbf{\eta}^\prime,\mathbf{\xi}^\prime) \right)$ for all $0 \leq t^\prime \leq t$} \\ + 0 & \mbox{otherwise} + \end{array}\right\}. + +where :math:`\mathbf{\eta}^\prime = \textbf{u}(\textbf{x}^{\prime +\prime},t) - \textbf{u}(\textbf{x}^\prime,t)` and +:math:`\mathbf{\xi}^\prime = \textbf{x}^{\prime \prime} - +\textbf{x}^\prime`. Here, :math:`s_0(t,\mathbf{\eta},\mathbf{\xi})` is a +critical stretch defined as + +.. math:: + + s_0(t,\mathbf{\eta},\mathbf{\xi}) = s_{00} - \alpha s_{\min}(t,\mathbf{\eta},\mathbf{\xi}), \qquad s_{\min}(t) = \min_{\mathbf{\xi}} s(t,\mathbf{\eta},\mathbf{\xi}), + +where :math:`s_{00}` and :math:`\alpha` are material-dependant +constants. The history function :math:`\mu` breaks bonds when the +stretch :math:`s` exceeds the critical stretch :math:`s_0`. + +Although :math:`s_0(t,\mathbf{\eta},\mathbf{\xi})` is expressed as a +property of a particle, bond breaking must be a symmetric operation for +all particle pairs sharing a bond. That is, particles :math:`\textbf{x}` +and :math:`\textbf{x}^\prime` must utilize the same test when deciding +to break their common bond. This can be done by any method that treats +the particles symmetrically. In the definition of :math:`\mu` above, we +have chosen to take the minimum of the two :math:`s_0` values for +particles :math:`\textbf{x}` and :math:`\textbf{x}^\prime` when +determining if the :math:`\textbf{x}`--:math:`\textbf{x}^\prime` bond +should be broken. + +Following :ref:`(Silling) `, we can define the damage at a +point :math:`\textbf{x}` as + +.. math:: + + \varphi(\textbf{x}, t) = 1 - \frac{\int_{\mathcal{H}_{\textbf{x}}} \mu(t,\mathbf{\eta},\mathbf{\xi}) dV_{\textbf{x}^\prime} + }{ \int_{\mathcal{H}_{\textbf{x}}} dV_{\textbf{x}^\prime} }. + +Discrete Peridynamic Model and LAMMPS Implementation +"""""""""""""""""""""""""""""""""""""""""""""""""""" + +In LAMMPS, instead of \eqref{eqn:NewtonII}, we model this equation of +motion: + +.. math:: + + \rho(\mathbf{x}) \ddot{\textbf{y}}(\mathbf{x},t) = \int_{\mathcal{H}_{\mathbf{x}}} + \left\{ \underline{\mathbf{T}}\left[ \mathbf{x},t \right]\left<\mathbf{x}^{\prime}-\mathbf{x} \right> + - \underline{\mathbf{T}}\left[\mathbf{x}^{\prime},t \right]\left<\mathbf{x}-\mathbf{x}^{\prime} \right> \right\} + {d}V_{\mathbf{x}^\prime} + \mathbf{b}(\mathbf{x},t), + +where we explicitly track and store at each timestep the positions and +not the displacements of the particles. We observe that +:math:`\ddot{\textbf{y}}(\textbf{x}, t) = \ddot{\textbf{x}} + +\ddot{\textbf{u}}(\textbf{x}, t) = \ddot{\textbf{u}}(\textbf{x}, t)`, so +that this is equivalent to \eqref{eqn:NewtonII}. + +Spatial Discretization +^^^^^^^^^^^^^^^^^^^^^^ + +The region defining a peridynamic material is discretized into particles +forming a simple cubic lattice with lattice constant :math:`\Delta x`, +where each particle :math:`i` is associated with some volume fraction +:math:`V_i`. For any particle :math:`i`, let :math:`\mathcal{F}_i` +denote the family of particles for which particle :math:`i` shares a +bond in the reference configuration. That is, + +.. math:: + + \mathcal{F}_i = \{ p ~ | ~ \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert \leq \delta \}. + +The discretized equation of motion replaces \eqref{eqn:NewtonII} with + +.. math:: + + \rho \ddot{\textbf{y}}_i^n = + \sum_{p \in \mathcal{F}_i} + \left\{ \underline{\mathbf{T}}\left[ \textbf{x}_i,t \right]\left<\textbf{x}_p^{\prime}-\textbf{x}_i \right> + - \underline{\mathbf{T}}\left[\textbf{x}_p,t \right]\left<\textbf{x}_i-\textbf{x}_p \right> \right\} + V_{p} + \textbf{b}_i^n, + +where :math:`n` is the timestep number and subscripts denote the particle number. + +Short-Range Forces +^^^^^^^^^^^^^^^^^^ + +In the model discussed so far, particles interact only through their +bond forces. A particle with no bonds becomes a free non-interacting +particle. To account for contact forces, short-range forces are +introduced :ref:`(Silling 2007) `. We add to the force in +\eqref{eqn:discreteNewtonII} the following force + +.. math:: + + \textbf{f}_S(\textbf{y}_p,\textbf{y}_i) = \min \left\{ 0, \frac{c_S}{\delta}(\left\Vert {\textbf{y}_p-\textbf{y}_i} \right\Vert - d_{pi}) \right\} + \frac{\textbf{y}_p-\textbf{y}_i}{\left\Vert {\textbf{y}_p-\textbf{y}_i} \right\Vert}, + +where :math:`d_{pi}` is the short-range interaction distance between +particles :math:`p` and :math:`i`, and :math:`c_S` is a multiple of the +constant :math:`c` from \eqref{eqn:c}. Note that the short-range force +is always repulsive, never attractive. In practice, we choose + +.. math:: + c_S = 15 \frac{18K}{\pi \delta^4}. + +For the short-range interaction distance, we choose :ref:`(Silling 2007) +` + +.. math:: + + d_{pi} = \min \left\{ 0.9 \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert, 1.35 (r_p + r_i) \right\}, + +where :math:`r_i` is called the *node radius* of particle +:math:`i`. Given a discrete lattice, we choose :math:`r_i` to be half +the lattice constant. + +.. note:: + + For a simple cubic lattice, :math:`\Delta x = \Delta y = \Delta z`. + +Given this definition of :math:`d_{pi}`, contact forces appear only when +particles are under compression. + +When accounting for short-range forces, it is convenient to define the +short-range family of particles + +.. math:: + + \mathcal{F}^S_i = \{ p ~ | ~ \left\Vert {\textbf{y}_p - \textbf{y}_i} \right\Vert \leq d_{pi} \}. + + +Modification to the Particle Volume +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The right-hand side of \eqref{eqn:discreteNewtonII} may be thought of as +a midpoint quadrature of \eqref{eqn:NewtonII}. To slightly improve the +accuracy of this quadrature, we discuss a modification to the particle +volume used in \eqref{eqn:discreteNewtonII}. In a situation where two +particles share a bond with :math:`\left\Vert { \textbf{x}_p - +\textbf{x}_i }\right\Vert = \delta`, for example, we suppose that only +approximately half the volume of each particle is "seen" by the other +:ref:`(Silling 2007) `. When computing the force of each +particle on the other we use :math:`V_p / 2` rather than :math:`V_p` in +\eqref{eqn:discreteNewtonII}. As such, we introduce a nodal volume +scaling function for all bonded particles where :math:`\delta - r_i \leq +\left\Vert { \textbf{x}_p - \textbf{x}_i } \right\Vert \leq \delta` (see +the Figure below). + +We choose to use a linear unitless nodal volume scaling function + +.. math:: + + \nu(\textbf{x}_p - \textbf{x}_i) = \left\{ + \begin{array}{cl} + -\frac{1}{2 r_i} \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert + \left( \frac{\delta}{2 r_i} + \frac{1}{2} \right) & \mbox{if } + \delta - r_i \leq \left\Vert {\textbf{x}_p - \textbf{x}_i } \right\Vert \leq \delta \\ + 1 & \mbox{if } \left\Vert {\textbf{x}_p - \textbf{x}_i } \right\Vert \leq \delta - r_i \\ + 0 & \mbox{otherwise} + \end{array} + \right\} + +If :math:`\left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert = \delta`, :math:`\nu = 0.5`, and if +:math:`\left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert = \delta - r_i`, :math:`\nu = 1.0`, for +example. + + +.. figure:: JPG/pdlammps_fig1.png + :figwidth: 80% + :figclass: align-center + + Diagram showing horizon of a particular particle, demonstrating that + the volume associated with particles near the boundary of the horizon is + not completely contained within the horizon. + +Temporal Discretization +^^^^^^^^^^^^^^^^^^^^^^^ + +When discretizing time in LAMMPS, we use a velocity-Verlet scheme, where +both the position and velocity of the particle are stored +explicitly. The velocity-Verlet scheme is generally expressed in three +steps. In :ref:`Algorithm 1 `, :math:`\rho_i` denotes the +mass density of a particle and :math:`\widetilde{\textbf{f}}_i^n` +denotes the the net force density on particle :math:`i` at timestep +:math:`n`. The LAMMPS command :doc:`fix nve ` performs a +velocity-Verlet integration. + + .. _algvelverlet: + + .. admonition:: Algorithm 1: Velocity Verlet + :class: tip + + | 1: :math:`\textbf{v}_i^{n + 1/2} = \textbf{v}_i^n + \frac{\Delta t}{2 \rho_i} \widetilde{\textbf{f}}_i^n` + | 2: :math:`\textbf{y}_i^{n+1} = \textbf{y}_i^n + \Delta t \textbf{v}_i^{n + 1/2}` + | 3: :math:`\textbf{v}_i^{n+1} = \textbf{v}_i^{n+1/2} + \frac{\Delta t}{2 \rho_i} \widetilde{\textbf{f}}_i^{n+1}` + +Breaking Bonds +^^^^^^^^^^^^^^ + +During the course of simulation, it may be necessary to break bonds, as +described in the :ref:`Damage section `. Bonds are recorded +as broken in a simulation by removing them from the bond family +:math:`\mathcal{F}_i` (see \eqref{eqn:BondFamily}). + +A naiive implementation would have us first loop over all bonds and +compute :math:`s_{min}` in \eqref{eqn:s0}, then loop over all bonds +again and break bonds with a stretch :math:`s > s0` as in +\eqref{eqn:mu}, and finally loop over all particles and compute forces +for the next step of :ref:`Algorithm 1 `. For reasons of +computational efficiency, we will utilize the values of :math:`s_0` from +the *previous* timestep when deciding to break a bond. + +.. note:: + + For the first timestep, :math:`s_0` is initialized to + :math:`\mathbf{\infty}` for all nodes. This means that no bonds may + be broken until the second timestep. As such, it is recommended that + the first few timesteps of the peridynamic simulation not involve any + actions that might result in the breaking of bonds. As a practical + example, the projectile in the :ref:`commented example below + ` is placed such that it does not impact the target + brittle plate until several timesteps into the simulation. + +LPS Pseudocode +^^^^^^^^^^^^^^ + +A sketch of the LPS model implementation in the PERI package appears in +:ref:`Algorithm 2 `. This algorithm makes use of the +routines in :ref:`Algorithm 3 ` and :ref:`Algorithm 4 +`. + + .. _algperilps: + + .. admonition:: Algorithm 2: LPS Peridynamic Model Pseudocode + :class: tip + + | Fix :math:`s_{00}`, :math:`\alpha`, horizon :math:`\delta`, bulk modulus :math:`K`, shear modulus :math:`G`, timestep :math:`\Delta t`, and generate initial lattice of particles with lattice constant :math:`\Delta x`. Let there be :math:`N` particles. Define constant :math:`c_S` for repulsive short-range forces. + | Initialize bonds between all particles :math:`i \neq j` where :math:`\left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert \leq \delta` + | Initialize weighted volume :math:`m` for all particles using :ref:`Algorithm 3 ` + | Initialize :math:`s_0 = \mathbf{\infty}` {*Initialize each entry to MAX_DOUBLE*} + | **while** not done **do** + | Perform step 1 of :ref:`Algorithm 1 `, updating velocities of all particles + | Perform step 2 of :ref:`Algorithm 1 `, updating positions of all particles + | :math:`\tilde{s}_0 = \mathbf{\infty}` {*Initialize each entry to MAX_DOUBLE*} + | **for** :math:`i=1` to :math:`N` **do** + | {Compute short-range forces} + | **for all** particles :math:`j \in \mathcal{F}^S_i` (the short-range family of nodes for particle :math:`i`) **do** + | :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert` + | :math:`dr = \min \{ 0, r - d \}` {*Short-range forces are only repulsive, never attractive*} + | :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in \eqref{eqn:cS}*} + | :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}` + | **end for** + | **end for** + | Compute the dilatation for each particle using :ref:`Algorithm 4 ` + | **for** :math:`i=1` to :math:`N` **do** + | {Compute bond forces} + | **for all** particles :math:`j` sharing an unbroken bond with particle :math:`i` **do** + | :math:`e = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert - \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert` + | :math:`\omega_+ = \underline{\omega}\left<\textbf{x}_j - \textbf{x}_i\right>` {*Influence function evaluation*} + | :math:`\omega_- = \underline{\omega}\left<\textbf{x}_i - \textbf{x}_j\right>` {*Influence function evaluation*} + | :math:`\hat{f} = \left[ (3K-5G)\left( \frac{\theta(i)}{m(i)}\omega_+ + \frac{\theta(j)}{m(j)}\omega_- \right) \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert + 15G \left( \frac{\omega_+}{m(i)} + \frac{\omega_-}{m(j)} \right) e \right] \nu(\textbf{x}_j - \textbf{x}_i) V_j` + | :math:`\textbf{f} = \textbf{f} + \hat{f} \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}` + | **if** :math:`(dr / \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert) > \min(s_0(i), s_0(j))` **then** + | Break :math:`i`'s bond with :math:`j` {:math:`j` *'s bond with* :math:`i` *will be broken when this loop iterates on* :math:`j`} + | **end if** + | :math:`\tilde{s}_0(i) = \min (\tilde{s}_0(i),s_{00}-\alpha(dr / \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert))` + | **end for** + | **end for** + | :math:`s_0 = \tilde{s}_0` {*Store for use in next timestep*} + | Perform step 3 of :ref:`Algorithm 1 `, updating velocities of all particles + | **end while** + + + .. _algperilpsm: + + .. admonition:: Algorithm 3: Computation of Weighted Volume *m* + :class: tip + + | **for** :math:`i=1` to :math:`N` **do** + | :math:`m(i) = 0.0` + | **for all** particles :math:`j` sharing a bond with particle :math:`i` **do** + | :math:`m(i) = m(i) + \underline{\omega}\left<\textbf{x}_j - \textbf{x}_i\right> \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert^2 \nu(\textbf{x}_j - \textbf{x}_i) V_j` + | **end for** + | **end for** + + .. _algperilpstheta: + + .. admonition:: Algorithm 4: Computation of Dilatation :math:`\theta` + :class: tip + + | **for** :math:`i=1` to :math:`N` **do** + | :math:`\theta(i) = 0.0` + | **for all** particles :math:`j` sharing an unbroken bond with particle :math:`i` **do** + | :math:`e = \left\Vert {\textbf{y}_i - \textbf{y}_j} \right\Vert - \left\Vert {\textbf{x}_i - \textbf{x}_j} \right\Vert` + | :math:`\theta(i) = \theta(i) + \underline{\omega}\left<\textbf{x}_j - \textbf{x}_i\right> \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert e \nu(\textbf{x}_j - \textbf{x}_i) V_j` + | **end for** + | :math:`\theta(i) = \frac{3}{m(i)}\theta(i)` + | **end for** + +PMB Pseudocode +^^^^^^^^^^^^^^ + +A sketch of the PMB model implementation in the PERI package appears in +:ref:`Algorithm 5 `. + + .. _algperipmb: + + .. admonition:: Algorithm 5: PMB Peridynamic Model Pseudocode + :class: tip + + | Fix :math:`s_{00}`, :math:`\alpha`, horizon :math:`\delta`, spring constant :math:`c`, timestep :math:`\Delta t`, and generate initial lattice of particles with lattice constant :math:`\Delta x`. Let there be :math:`N` particles. + | Initialize bonds between all particles :math:`i \neq j` where :math:`\left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert \leq \delta` + | Initialize :math:`s_0 = \mathbf{\infty}` {*Initialize each entry to MAX_DOUBLE*} + | **while** not done **do** + | Perform step 1 of :ref:`Algorithm 1 `, updating velocities of all particles + | Perform step 2 of :ref:`Algorithm 1 `, updating positions of all particles + | :math:`\tilde{s}_0 = \mathbf{\infty}` {*Initialize each entry to MAX_DOUBLE*} + | **for** :math:`i=1` to :math:`N` **do** + | {Compute short-range forces} + | **for all** particles :math:`j \in \mathcal{F}^S_i` (the short-range family of nodes for particle :math:`i`) **do** + | :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert` + | :math:`dr = \min \{ 0, r - d \}` {*Short-range forces are only repulsive, never attractive*} + | :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in \eqref{eqn:cS}*} + | :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}` + | **end for** + | **end for** + | **for** :math:`i=1` to :math:`N` **do** + | {Compute bond forces} + | **for all** particles :math:`j` sharing an unbroken bond with particle :math:`i` **do** + | :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert` + | :math:`dr = r - \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert` + | :math:`k = \frac{c}{\left\Vert {\textbf{x}_i - \textbf{x}_j} \right\Vert} \nu(\textbf{x}_i - \textbf{x}_j) V_j dr` {:math:`c` *defined in \eqref{eqn:c}*} + | :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}` + | **if** :math:`(dr / \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert) > \min(s_0(i), s_0(j))` **then** + | Break :math:`i`'s bond with :math:`j` {:math:`j`\ *'s bond with* :math:`i` *will be broken when this loop iterates on* :math:`j`} + | **end if** + | :math:`\tilde{s}_0(i) = \min (\tilde{s}_0(i),s_{00}-\alpha(dr / \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert))` + | **end for** + | **end for** + | :math:`s_0 = \tilde{s}_0` {*Store for use in next timestep*} + | Perform step 3 of :ref:`Algorithm 1 `, updating velocities of all particles + | **end while** + +Damage +^^^^^^ + +The damage associated with every particle (see \eqref{eqn:damage}) can +optionally be computed and output with a LAMMPS data dump. To do this, +your input script must contain the command :doc:`compute damage/atom +` This enables a LAMMPS per-atom compute to +calculate the damage associated with each particle every time a LAMMPS +:doc:`data dump ` frame is written. + +Visualizing Simulation Results +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +There are multiple ways to visualize the simulation results. Typically, +you want to display the particles and color code them by the value +computed with the :doc:`compute damage/atom ` +command. + +This can be done, for example, by using the built-in visualizer of the +:doc:`dump image or dump movie ` command to create snapshot +images or a movie. Below are example command lines for using dump image +with the :ref:`example listed below ` and a set of images +created for steps 300, 600, and 2000 this way. + +.. code-block:: LAMMPS + + dump D2 all image 100 dump.peri.*.png c_C1 type box no 0.0 view 30 60 zoom 1.5 up 0 0 -1 ssao yes 4539 0.6 + dump_modify D2 pad 5 adiam * 0.001 amap 0.0 1.0 ca 0.1 3 min blue 0.5 yellow max red + +.. |periimage1| image:: JPG/dump.peri.300.png + :width: 32% + +.. |periimage2| image:: JPG/dump.peri.600.png + :width: 32% + +.. |periimage3| image:: JPG/dump.peri.2000.png + :width: 32% + +|periimage1| |periimage2| |periimage3| + +For interactive visualization, the `Ovito `_ is very +convenient to use. Below are steps to create a visualization of the +:ref:`same example from below ` now using the generated +trajectory in the ``dump.peri`` file. + +- Launch Ovito +- File -> Load File -> ``dump.peri`` +- Select "-> Particle types" and under "Appearance" set "Display radius:" to 0.0005 +- From the "Add modification:" drop down list select "Color coding" +- Under "Color coding" select from the "Color gradient" drop down list "Jet" +- Also under "Color coding" set "Start value:" to 0 and "End value:" to 1 +- You can improve the image quality by adding the "Ambient occlusion" modification + +.. figure:: JPG/ovito-peri-snap.png + :figwidth: 80% + :figclass: align-center + + Screenshot of visualizing a trajectory with Ovito + +Pitfalls +^^^^^^^^ + +**Parallel Scalability** + +LAMMPS operates in parallel in a :doc:`spatial-decomposition mode +`, where each processor owns a spatial subdomain of +the overall simulation domain and communicates with its neighboring +processors via distributed-memory message passing (MPI) to acquire ghost +atom information to allow forces on the atoms it owns to be +computed. LAMMPS also uses Verlet neighbor lists which are recomputed +every few timesteps as particles move. On these timesteps, particles +also migrate to new processors as needed. LAMMPS decomposes the overall +simulation domain so that spatial subdomains of nearly equal volume are +assigned to each processor. When each subdomain contains nearly the same +number of particles, this results in a reasonable load balance among all +processors. As is more typical with some peridynamic simulations, some +subdomains may contain many particles while other subdomains contain few +particles, resulting in a load imbalance that impacts parallel +scalability. + +**Setting the "skin" distance** + +The :doc:`neighbor ` command with LAMMPS is used to set the +so-called "skin" distance used when building neighbor lists. All atom +pairs within a cutoff distance equal to the horizon :math:`\delta` plus +the skin distance are stored in the list. Unexpected crashes in LAMMPS +may be due to too small a skin distance. The skin should be set to a +value such that :math:`\delta` plus the skin distance is larger than the +maximum possible distance between two bonded particles. For example, if +:math:`s_{00}` is increased, the skin distance may also need to be +increased. + +**"Lost" particles** + +All particles are contained within the "simulation box" of LAMMPS. The +boundaries of this box may change with time, or not, depending on how +the LAMMPS :doc:`boundary ` command has been set. If a +particle drifts outside the simulation box during the course of a +simulation, it is called *lost*. + +As an option of the :doc:`themo_modify ` command of +LAMMPS, the lost keyword determines whether LAMMPS checks for lost atoms +each time it computes thermodynamics and what it does if atoms are +lost. If the value is *ignore*, LAMMPS does not check for lost atoms. If +the value is *error* or *warn*, LAMMPS checks and either issues an error +or warning. The code will exit with an error and continue with a +warning. This can be a useful debugging option. The default behavior of +LAMMPS is to exit with an error if a particle is lost. + +The peridynamic module within LAMMPS does not check for lost atoms. If a +particle with unbroken bonds is lost, those bonds are marked as broken +by the remaining particles. + +**Defining the peridynamic horizon** :math:`\mathbf{\delta}` + +In the :doc:`pair_coeff ` command, the user must specify the +horizon :math:`\delta`. This argument determines which particles are +bonded when the simulation is initialized. It is recommended that +:math:`\delta` be set to a small fraction of a lattice constant larger than +desired. + +For example, if the lattice constant is 0.0005 and you wish to set the +horizon to three times the lattice constant, then set :math:`\delta` to +be 0.0015001, a value slightly larger than three times the lattice +constant. This guarantees that particles three lattice constants away +from each other are still bonded. If :math:`\delta` is set to 0.0015, +for example, floating point error may result in some pairs of particles +three lattice constants apart not being bonded. + +**Breaking bonds too early** + +For technical reasons, the bonds in the simulation are not created until +the end of the first timestep of the simulation. Therefore, one should +not attempt to break bonds until at least the second step of the +simulation. + +Bugs +^^^^ + +The user is cautioned that this code is a beta release. If you are +confident that you have found a bug in the peridynamic module, please +report it in a `GitHub Issue ` +or send an email to the LAMMPS developers. First, check the `New +features and bug fixes `_ section of +the LAMMPS website site to see if the bug has already been reported or +fixed. If not, the most useful thing you can do for us is to isolate the +problem. Run it on the smallest number of atoms and fewest number of +processors and with the simplest input script that reproduces the +bug. In your message, describe the problem and any ideas you have as to +what is causing it or where in the code the problem might be. We'll +request your input script and data files if necessary. + +Modifying and Extending the Peridynamic Module +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +To add new features or peridynamic potentials to the peridynamic module, +the user is referred to the :doc:`Modifying & extending LAMMPS ` +section. To develop a new bond-based material, start with the *peri/pmb* +pair style as a template. To develop a new state-based material, start +with the *peri/lps* pair style as a template. + +A Numerical Example +""""""""""""""""""" + +To introduce the peridynamic implementation within LAMMPS, we replicate +a numerical experiment taken from section 6 of :ref:`(Silling 2005) +`. + +Problem Description and Setup +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +We consider the impact of a rigid sphere on a homogeneous disk of +brittle material. The sphere has diameter :math:`0.01` m and velocity +100 m/s directed normal to the surface of the target. The target +material has density :math:`\rho = 2200` kg/m:math:`^3`. A PMB material +model is used with :math:`K = 14.9` GPa and critical bond stretch +parameters given by :math:`s_{00} = 0.0005` and :math:`\alpha = 0.25`. A +three-dimensional simple cubic lattice is constructed with lattice +constant :math:`0.0005` m and horizon :math:`0.0015` m. (The horizon is +three times the lattice constant.) The target is a cylinder of diameter +:math:`0.074` m and thickness :math:`0.0025` m, and the associated +lattice contains 103,110 particles. Each particle :math:`i` has volume +fraction :math:`V_i = 1.25 \times 10^{-10} \textrm{m}^3`. + +The spring constant in the PMB material model is (see \eqref{eqn:c}) + +.. math:: + c = \frac{18k}{\pi \delta^4} = \frac{18 (14.9 \times 10^9)}{ \pi (1.5 \times 10^{-3})^4} \approx 1.6863 \times 10^{22}. + +The CFL analysis from :ref:`(Silling2005) ` shows that a +timestep of :math:`1.0 \times 10^{-7}` is safe. + +We observe here that in IEEE double-precision floating point arithmetic +when computing the bond stretch :math:`s(t,\mathbf{\eta},\mathbf{\xi})` +at each iteration where :math:`\left\Vert {\mathbf{\eta}+\mathbf{\xi}} +\right\Vert` is computed during the iteration and :math:`\left\Vert +{\mathbf{\xi}} \right\Vert` was computed and stored for the initial +lattice, it may be that :math:`fl(s) = \varepsilon` with :math:`\left| +\varepsilon \right| \leq \varepsilon_{machine}` for an unstretched +bond. Taking :math:`\varepsilon = 2.220446049250313 \times 10^{-16}`, we +see that the value :math:`c s V_i \approx 4.68 \times 10^{-4}`, computed +when determining :math:`f`, is perhaps larger than we would like, +especially when the true force should be zero. One simple way to avoid +this issue is to insert the following instructions in Algorithm +:ref:`Algorithm 5 ` after instruction 21 (and similarly for +Algorithm :ref:`Algorithm 2 `): + + | **if** :math:`\left| dr \right| < \varepsilon_{machine}` **then** + | :math:`dr = 0` + | **end if** + +Qualitatively, this says that displacements from equilibrium on the +order of :math:`10^{-16}`\ m are taken to be exactly zero, a seemingly +reasonable assumption. + +The Projectile +^^^^^^^^^^^^^^ + +The projectile used in the following experiments is not the one used in +:ref:`(Silling 2005) `. The projectile used here exerts a +force + +.. math:: + + F(r) = - k_s (r - R)^2 + +on each atom where :math:`k_s` is a specified force constant, :math:`r` is +the distance from the atom to the center of the indenter, and :math:`R` +is the radius of the projectile. The force is repulsive and :math:`F(r) = +0` for :math:`r > R`. For our problem, the projectile radius :math:`R = +0.05` m, and we have chosen :math:`k_s = 1.0 \times 10^{17}` (compare +with \eqref{eqn:c} above). + +Writing the LAMMPS Input File +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +We discuss the example input script :ref:`listed below `. +In line 2 we specify that SI units are to be used. We specify the +dimension (3) and boundary conditions ("shrink-wrapped") for the +computational domain in lines 3 and 4. In line 5 we specify that +peridynamic particles are to be used for this simulation. In line 7, we +set the "skin" distance used in building the LAMMPS neighborlist. In +line 8 we set the lattice constant (in meters) and in line 10 we define +the spatial region where the target will be placed. In line 12 we +specify a rectangular box enclosing the target region that defines the +simulation domain. Line 14 fills the target region with atoms. Lines 15 +and 17 define the peridynamic material model, and lines 19 and 21 set +the particle density and particle volume, respectively. The particle +volume should be set to the cube of the lattice constant for a simple +cubic lattice. Line 23 sets the initial velocity of all particles to +zero. Line 25 instructs LAMMPS to integrate time with velocity-Verlet, +and lines 27-30 create the spherical projectile, sending it with a +velocity of 100 m/s towards the target. Line 32 declares a compute style +for the damage (percentage of broken bonds) associated with each +particle. Line 33 sets the timestep, line 34 instructs LAMMPS to +provide a screen dump of thermodynamic quantities every 200 timesteps, +and line 35 instructs LAMMPS to create a data file (``dump.peri``) with +a complete snapshot of the system every 100 timesteps. This file can be +used to create still images or movies. Finally, line 36 instructs LAMMPS +to run for 2000 timesteps. + +.. _periexample: + +.. code-block:: LAMMPS + :linenos: + :caption: Peridynamics Example LAMMPS Input Script + + # 3D Peridynamic simulation with projectile" + units si + dimension 3 + boundary s s s + atom_style peri + atom_modify map array + neighbor 0.0010 bin + lattice sc 0.0005 + # Create desired target + region target cylinder y 0.0 0.0 0.037 -0.0025 0.0 units box + # Make 1 atom type + create_box 1 target + # Create the atoms in the simulation region + create_atoms 1 region target + pair_style peri/pmb + # + pair_coeff * * 1.6863e22 0.0015001 0.0005 0.25 + # Set mass density + set group all density 2200 + # volume = lattice constant^3 + set group all volume 1.25e-10 + # Zero out velocities of particles + velocity all set 0.0 0.0 0.0 sum no units box + # Use velocity-Verlet time integrator + fix F1 all nve + # Construct spherical indenter to shatter target + variable y0 equal 0.00510 + variable vy equal -100 + variable y equal "v_y0 + step*dt*v_vy" + fix F2 all indent 1e17 sphere 0.0000 v_y 0.0000 0.0050 units box + # Compute damage for each particle + compute C1 all damage/atom + timestep 1.0e-7 + thermo 200 + dump D1 all custom 100 dump.peri id type x y z c_C1 + run 2000 + +.. note:: + + To use the LPS model, replace line 15 with :doc:`pair_style peri/lps + ` and modify line 16 accordingly. + +Numerical Results and Discussion +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +We ran the :ref:`input script from above `. Images of the +disk (projectile not shown) appear in Figure below. The plot of damage +on the top monolayer was created by coloring each particle according to +its damage. + +The symmetry in the computed solution arises because a "perfect" lattice +was used, and a because a perfectly spherical projectile impacted the +lattice at its geometric center. To break the symmetry in the solution, +the nodes in the peridynamic body may be perturbed slightly from the +lattice sites. To do this, the lattice of points can be slightly +perturbed using the :doc:`displace_atoms ` command. + +.. _figperitarget: + +.. figure:: JPG/pdlammps_fig2.png + :figwidth: 80% + :figclass: align-center + + Target during (a) and after (b,c) impact + +------------ + +.. _Emmrich2007: + +**(Emmrich)** Emmrich, Weckner, Commun. Math. Sci., 5, 851-864 (2007), + +.. _Parks2: + +**(Parks)** Parks, Lehoucq, Plimpton, Silling, Comp Phys Comm, 179(11), 777-783 (2008). + +.. _Silling2000_2: + +**(Silling 2000)** Silling, J Mech Phys Solids, 48, 175-209 (2000). + +.. _Silling2005: + +**(Silling 2005)** Silling Askari, Computer and Structures, 83, 1526-1535 (2005). + +.. _Silling2007_2: + +**(Silling 2007)** Silling, Epton, Weckner, Xu, Askari, J Elasticity, 88, 151-184 (2007). + +.. _Seleson2010: + +**(Seleson 2010)** Seleson, Parks, Int J Mult Comp Eng 9(6), pp. 689-706, 2011. diff --git a/doc/src/JPG/dump.peri.2000.png b/doc/src/JPG/dump.peri.2000.png new file mode 100644 index 0000000000000000000000000000000000000000..d8cacb4bb2c7d205925798545939924b355b17d7 GIT binary patch literal 205139 zcmeFZRaYEc7cEQ@k^sRWxYM}1LvW|lIE@8&cYo4YaA<-<6B>7S3lQAhA<(!6clda} zG0u-TW1PC#bx{|!Ywf+(+;h%V5nt8hG0;fRkdTlt6cuFDk&xcL-o8ab`QPA2B3=GE zkh;t2x@$ODx_g?$ak({u&NIRpA-Wa)p8_`aR~*nJU(QKI=^;r)%VCA@s)yI;8$9@bZ$8J9 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zJA-o8OMJaCIpYOsn^r`~hu#1bh@nevk)wzo6^ zULLzqD^FS3X%*a(phpkc=hF>_<%o)x;?{r8zS%WPbI{#F@5c1nS$QzGlXjHNClPW4 z))0A9nJbl#Lehw-1_9bl|CM>3w)Z2VCM9%9sZ|J(u}mL)UZbpuoirhGIf2;5jTa-r zDrBkUf=EgDFWfrRdyn9^oXoNFU9XT`E7bF4!wzUt&=I7W%qMe+DBk@sNVt{dtL`x- zy^a#7B+&7yhy~7HncN=`bW zKNeC#6jL#Q*k3WFIcb10l?pM6l1snJ8Dl(OZE>7$}8CS+?dNsr)KV@1h zCii(CCYpD4!CE0|OrLrcIxhwSk`qn^I#E@&lixrDuhj`(93n7Sl?oMfY^c=@1YPWd|&{9ork;r>ijlclak;cuiN=t-k6v4Jx zi#<-b)eWkgzU@u?KbuN~9VT zrGY3<0ge6TCCy7bE%7I`Tm?>J`hYlM$;kFa`Co zoaw?E-7PCNv_2`{mp6`|omEGDn?QXf)uiQ9^qO!r-*RT6=>ZWpU)ueMyAzK|;0ial zVY`LY7)wJC{ow6Qb}rR45=Hp;tEhY61Dbra0hkK_r~pec`@)w|h6DeiCo_Wt=uGNvz!hw$aW6Cw0PURA;NOp`{tiy=Ly?Jj6dV3 zX!l?9(SEyj{=uxQS+B&!B!4#i?~5_}TJS&a#_j)KPKm{x|L-T>|M$Knq<7HZlw!x& T?myGT#Top(V#&3|zwG~iQW^p! literal 0 HcmV?d00001 From 8e073c3335785de520b7b31a963286b205461ff6 Mon Sep 17 00:00:00 2001 From: Axel Kohlmeyer Date: Thu, 29 Sep 2022 07:43:01 -0400 Subject: [PATCH 2/4] reference the Howto from the rest of the manual. Spelling fixes. --- doc/src/Howto_peri.rst | 38 +++++++-------- doc/src/Packages_details.rst | 1 + doc/src/compute_damage_atom.rst | 16 ++++--- doc/src/compute_dilatation_atom.rst | 20 ++++---- doc/src/pair_peri.rst | 52 +++++++++++---------- doc/utils/sphinx-config/false_positives.txt | 10 +++- 6 files changed, 77 insertions(+), 60 deletions(-) diff --git a/doc/src/Howto_peri.rst b/doc/src/Howto_peri.rst index 6285036fc8..99586da888 100644 --- a/doc/src/Howto_peri.rst +++ b/doc/src/Howto_peri.rst @@ -16,8 +16,8 @@ model is discretized within LAMMPS as described in the original article :ref:`(Parks) `. An example problem with comments is also included. -Quickstart -"""""""""" +Quick Start +""""""""""" The peridynamics styles are included in the optional :ref:`PERI package `. If your LAMMPS executable does not already include the @@ -54,7 +54,7 @@ Some notes on this input example: - particles must be created on a :doc:`simple cubic lattice ` - using the :doc:`atom style peri ` is required - an :doc:`atom map ` is required for indexing particles -- The :doc:`skin distance ` used when computing neighborlists +- The :doc:`skin distance ` used when computing neighbor lists should be defined appropriately for your choice of simulation parameters. The *skin* should be set to a value such that the peridynamic horizon plus the skin distance is larger than the maximum @@ -254,11 +254,11 @@ extension :math:`\underline{e}^{\rm d}`. not change as bonds break. It is computed with respect to the bond family defined at the reference (initial) configuration. -The nonnegative scalar state :math:`\underline{\omega}` is an *influence -function* :ref:`(Silling 2007) `. For more on influence -functions, see :ref:`(Seleson 2010) `. If an influence -function :math:`\underline{\omega}` depends only upon the scalar -:math:`\left\Vert \boldsymbol{\xi} \right\Vert`, (i.e., +The non-negative scalar state :math:`\underline{\omega}` is an +*influence function* :ref:`(Silling 2007) `. For more on +influence functions, see :ref:`(Seleson 2010) `. If an +influence function :math:`\underline{\omega}` depends only upon the +scalar :math:`\left\Vert \boldsymbol{\xi} \right\Vert`, (i.e., :math:`\underline{\omega}\left<\boldsymbol{\xi}\right> = \underline{\omega}\left<\left\Vert \boldsymbol{\xi} \right\Vert\right>`\ ), then :math:`\underline{\omega}` is a spherical influence function. @@ -369,7 +369,7 @@ critical stretch defined as s_0(t,\mathbf{\eta},\mathbf{\xi}) = s_{00} - \alpha s_{\min}(t,\mathbf{\eta},\mathbf{\xi}), \qquad s_{\min}(t) = \min_{\mathbf{\xi}} s(t,\mathbf{\eta},\mathbf{\xi}), -where :math:`s_{00}` and :math:`\alpha` are material-dependant +where :math:`s_{00}` and :math:`\alpha` are material-dependent constants. The history function :math:`\mu` breaks bonds when the stretch :math:`s` exceeds the critical stretch :math:`s_0`. @@ -557,7 +557,7 @@ described in the :ref:`Damage section `. Bonds are recorded as broken in a simulation by removing them from the bond family :math:`\mathcal{F}_i` (see \eqref{eqn:BondFamily}). -A naiive implementation would have us first loop over all bonds and +A naive implementation would have us first loop over all bonds and compute :math:`s_{min}` in \eqref{eqn:s0}, then loop over all bonds again and break bonds with a stretch :math:`s > s0` as in \eqref{eqn:mu}, and finally loop over all particles and compute forces @@ -761,20 +761,20 @@ Pitfalls **Parallel Scalability** LAMMPS operates in parallel in a :doc:`spatial-decomposition mode -`, where each processor owns a spatial subdomain of +`, where each processor owns a spatial sub-domain of the overall simulation domain and communicates with its neighboring processors via distributed-memory message passing (MPI) to acquire ghost atom information to allow forces on the atoms it owns to be computed. LAMMPS also uses Verlet neighbor lists which are recomputed every few timesteps as particles move. On these timesteps, particles also migrate to new processors as needed. LAMMPS decomposes the overall -simulation domain so that spatial subdomains of nearly equal volume are -assigned to each processor. When each subdomain contains nearly the same -number of particles, this results in a reasonable load balance among all -processors. As is more typical with some peridynamic simulations, some -subdomains may contain many particles while other subdomains contain few -particles, resulting in a load imbalance that impacts parallel -scalability. +simulation domain so that spatial sub-domains of nearly equal volume are +assigned to each processor. When each sub-domain contains nearly the +same number of particles, this results in a reasonable load balance +among all processors. As is more typical with some peridynamic +simulations, some sub-domains may contain many particles while other +sub-domains contain few particles, resulting in a load imbalance that +impacts parallel scalability. **Setting the "skin" distance** @@ -937,7 +937,7 @@ In line 2 we specify that SI units are to be used. We specify the dimension (3) and boundary conditions ("shrink-wrapped") for the computational domain in lines 3 and 4. In line 5 we specify that peridynamic particles are to be used for this simulation. In line 7, we -set the "skin" distance used in building the LAMMPS neighborlist. In +set the "skin" distance used in building the LAMMPS neighbor list. In line 8 we set the lattice constant (in meters) and in line 10 we define the spatial region where the target will be placed. In line 12 we specify a rectangular box enclosing the target region that defines the diff --git a/doc/src/Packages_details.rst b/doc/src/Packages_details.rst index f89ab00039..64037e950f 100644 --- a/doc/src/Packages_details.rst +++ b/doc/src/Packages_details.rst @@ -2200,6 +2200,7 @@ Foster (UTSA). **Supporting info:** * src/PERI: filenames -> commands +* :doc:`Peridynamics Howto ` * `doc/PDF/PDLammps_overview.pdf `_ * `doc/PDF/PDLammps_EPS.pdf `_ * `doc/PDF/PDLammps_VES.pdf `_ diff --git a/doc/src/compute_damage_atom.rst b/doc/src/compute_damage_atom.rst index b75a3ebc57..d396fe7424 100644 --- a/doc/src/compute_damage_atom.rst +++ b/doc/src/compute_damage_atom.rst @@ -24,16 +24,17 @@ Description """"""""""" Define a computation that calculates the per-atom damage for each atom -in a group. This is a quantity relevant for :doc:`Peridynamics models `. See `this document `_ -for an overview of LAMMPS commands for Peridynamics modeling. +in a group. This is a quantity relevant for :doc:`Peridynamics models +`. See `this document `_ for an +overview of LAMMPS commands for Peridynamics modeling. The "damage" of a Peridynamics particles is based on the bond breakage between the particle and its neighbors. If all the bonds are broken the particle is considered to be fully damaged. -See the `PDLAMMPS user guide `_ for a formal -definition of "damage" and more details about Peridynamics as it is -implemented in LAMMPS. +See the :doc:`Peridynamics Howto ` for a formal definition +of "damage" and more details about Peridynamics as it is implemented in +LAMMPS. This command can be used with all the Peridynamic pair styles. @@ -53,8 +54,9 @@ The per-atom vector values are unitless numbers (damage) :math:`\ge 0.0`. Restrictions """""""""""" -This compute is part of the PERI package. It is only enabled if -LAMMPS was built with that package. See the :doc:`Build package ` page for more info. +This compute is part of the PERI package. It is only enabled if LAMMPS +was built with that package. See the :doc:`Build package +` page for more info. Related commands """""""""""""""" diff --git a/doc/src/compute_dilatation_atom.rst b/doc/src/compute_dilatation_atom.rst index 8e3c86a4af..6ccc65dec3 100644 --- a/doc/src/compute_dilatation_atom.rst +++ b/doc/src/compute_dilatation_atom.rst @@ -24,7 +24,8 @@ Description """"""""""" Define a computation that calculates the per-atom dilatation for each -atom in a group. This is a quantity relevant for :doc:`Peridynamics models `. See `this document `_ +atom in a group. This is a quantity relevant for :doc:`Peridynamics +models `. See `this document `_ for an overview of LAMMPS commands for Peridynamics modeling. For small deformation, dilatation of is the measure of the volumetric @@ -32,13 +33,14 @@ strain. The dilatation :math:`\theta` for each peridynamic particle :math:`i` is calculated as a sum over its neighbors with unbroken bonds, where the -contribution of the :math:`ij` pair is a function of the change in bond length -(versus the initial length in the reference state), the volume +contribution of the :math:`ij` pair is a function of the change in bond +length (versus the initial length in the reference state), the volume fraction of the particles and an influence function. See the -`PDLAMMPS user guide `_ for -a formal definition of dilatation. +:doc:`Peridynamics Howto ` for a formal definition of +dilatation. -This command can only be used with a subset of the Peridynamic :doc:`pair styles `: peri/lps, peri/ves and peri/eps. +This command can only be used with a subset of the Peridynamic +:doc:`pair styles `: *peri/lps*, *peri/ves*, and *peri/eps*. The dilatation value will be 0.0 for atoms not in the specified compute group. @@ -56,9 +58,9 @@ The per-atom vector values are unitless numbers :math:`(\theta \ge 0.0)`. Restrictions """""""""""" -This compute is part of the PERI package. It is only enabled if -LAMMPS was built with that package. See the -:doc:`Build package ` page for more info. +This compute is part of the PERI package. It is only enabled if LAMMPS +was built with that package. See the :doc:`Build package +` page for more info. Related commands """""""""""""""" diff --git a/doc/src/pair_peri.rst b/doc/src/pair_peri.rst index 45b406dd61..1fb94acd37 100644 --- a/doc/src/pair_peri.rst +++ b/doc/src/pair_peri.rst @@ -51,8 +51,9 @@ Description """"""""""" The peridynamic pair styles implement material models that can be used -at the mesoscopic and macroscopic scales. See `this document `_ for an overview of LAMMPS commands -for Peridynamics modeling. +at the mesoscopic and macroscopic scales. See `this document +`_ for an overview of LAMMPS commands for +Peridynamics modeling. Style *peri/pmb* implements the Peridynamic bond-based prototype microelastic brittle (PMB) model. @@ -66,26 +67,27 @@ peridynamic viscoelastic solid (VES) model. Style *peri/eps* implements the Peridynamic state-based elastic-plastic solid (EPS) model. -The canonical papers on Peridynamics are :ref:`(Silling 2000) ` -and :ref:`(Silling 2007) `. The implementation of Peridynamics -in LAMMPS is described in :ref:`(Parks) `. Also see the `PDLAMMPS user guide `_ for -more details about its implementation. +The canonical papers on Peridynamics are :ref:`(Silling 2000) +` and :ref:`(Silling 2007) `. The +implementation of Peridynamics in LAMMPS is described in :ref:`(Parks) +`. Also see the :doc:`Peridynamics Howto ` for more +details about its implementation. The peridynamic VES and EPS models in PDLAMMPS were implemented by R. Rahman and J. T. Foster at University of Texas at San Antonio. The original VES formulation is described in "(Mitchell2011)" and the original EPS formulation is in "(Mitchell2011a)". Additional PDF docs -that describe the VES and EPS implementations are include in the -LAMMPS distribution in `doc/PDF/PDLammps_VES.pdf `_ and +that describe the VES and EPS implementations are include in the LAMMPS +distribution in `doc/PDF/PDLammps_VES.pdf `_ and `doc/PDF/PDLammps_EPS.pdf `_. For questions regarding the VES and EPS models in LAMMPS you can contact R. Rahman (rezwanur.rahman at utsa.edu). The following coefficients must be defined for each pair of atom types via the :doc:`pair_coeff ` command as in the examples above, -or in the data file or restart files read by the -:doc:`read_data ` or :doc:`read_restart ` -commands, or by mixing as described below. +or in the data file or restart files read by the :doc:`read_data +` or :doc:`read_restart ` commands, or by +mixing as described below. For the *peri/pmb* style: @@ -96,8 +98,8 @@ For the *peri/pmb* style: C is the effectively a spring constant for Peridynamic bonds, the horizon is a cutoff distance for truncating interactions, and s00 and -:math:`\alpha` are used as a bond breaking criteria. The units of c are such -that c/distance = stiffness/volume\^2, where stiffness is +:math:`\alpha` are used as a bond breaking criteria. The units of c are +such that c/distance = stiffness/volume\^2, where stiffness is energy/distance\^2 and volume is distance\^3. See the users guide for more details. @@ -110,8 +112,8 @@ For the *peri/lps* style: * :math:`\alpha` (unitless) K is the bulk modulus and G is the shear modulus. The horizon is a -cutoff distance for truncating interactions, and s00 and :math:`\alpha` are -used as a bond breaking criteria. See the users guide for more +cutoff distance for truncating interactions, and s00 and :math:`\alpha` +are used as a bond breaking criteria. See the users guide for more details. For the *peri/ves* style: @@ -125,12 +127,12 @@ For the *peri/ves* style: * m_taubi (unitless) K is the bulk modulus and G is the shear modulus. The horizon is a -cutoff distance for truncating interactions, and s00 and :math:`\alpha` are -used as a bond breaking criteria. m_lambdai and m_taubi are the +cutoff distance for truncating interactions, and s00 and :math:`\alpha` +are used as a bond breaking criteria. m_lambdai and m_taubi are the viscoelastic relaxation parameter and time constant, -respectively. m_lambdai varies within zero to one. For very small -values of m_lambdai the viscoelastic model responds very similar to a -linear elastic model. For details please see the description in +respectively. m_lambdai varies within zero to one. For very small values +of m_lambdai the viscoelastic model responds very similar to a linear +elastic model. For details please see the description in "(Mitchell2011)". For the *peri/eps* style: @@ -165,8 +167,9 @@ shift option. The :doc:`pair_modify ` table and tail options are not relevant for these pair styles. -These pair styles write their information to :doc:`binary restart files `, so pair_style and pair_coeff commands do not need -to be specified in an input script that reads a restart file. +These pair styles write their information to :doc:`binary restart files +`, so pair_style and pair_coeff commands do not need to be +specified in an input script that reads a restart file. These pair styles can only be used via the *pair* keyword of the :doc:`run_style respa ` command. They do not support the @@ -177,8 +180,9 @@ These pair styles can only be used via the *pair* keyword of the Restrictions """""""""""" -All of these styles are part of the PERI package. They are only -enabled if LAMMPS was built with that package. See the :doc:`Build package ` page for more info. +All of these styles are part of the PERI package. They are only enabled +if LAMMPS was built with that package. See the :doc:`Build package +` page for more info. Related commands """""""""""""""" diff --git a/doc/utils/sphinx-config/false_positives.txt b/doc/utils/sphinx-config/false_positives.txt index c719620fc6..78d321ec50 100644 --- a/doc/utils/sphinx-config/false_positives.txt +++ b/doc/utils/sphinx-config/false_positives.txt @@ -747,6 +747,7 @@ dirname discoverable discretization discretized +discretizing disp dissipative Dissipative @@ -915,6 +916,7 @@ emax Emax Embt emi +Emmrich emol eN endian @@ -2125,6 +2127,7 @@ modelled modelling Modelling Modine +moduli mofff MOFFF Mohd @@ -2140,6 +2143,7 @@ Monaghan Monaghans monodisperse monodispersity +monolayer monopole monovalent Montalenti @@ -2415,6 +2419,7 @@ normy normz Noskov noslip +notational noticable Nout noutcol @@ -2775,6 +2780,7 @@ ps Ps pscreen pscrozi +Pseudocode pseudodynamics pseudopotential pSp @@ -3134,6 +3140,7 @@ sectoring sed segmental Seifert +Seleson sellerio Sellerio Semaev @@ -3212,7 +3219,7 @@ slategray slater Slepoy Sliozberg -sLLG +sLL sllod sm smallbig @@ -3623,6 +3630,7 @@ unsmoothed unsolvated unsplit unstrained +unstretched untar untilted Unwin From c6eb6d858b7a62d7a7344fcef776b36c3b273786 Mon Sep 17 00:00:00 2001 From: Axel Kohlmeyer Date: Thu, 29 Sep 2022 12:04:43 -0400 Subject: [PATCH 3/4] implement numbered equations and equation references --- doc/src/Howto_peri.rst | 70 ++++++++++++------- .../sphinx-config/_static/css/lammps.css | 8 +++ 2 files changed, 53 insertions(+), 25 deletions(-) diff --git a/doc/src/Howto_peri.rst b/doc/src/Howto_peri.rst index 99586da888..ff303f16aa 100644 --- a/doc/src/Howto_peri.rst +++ b/doc/src/Howto_peri.rst @@ -130,13 +130,15 @@ notation of :ref:`(Silling 2007) `, we write the peridynamic equation of motion as .. math:: - + :label: periNewtonII \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x},t) = - \int_{\mathcal{H}_{\mathbf{x}}} \left\{ \underline{\mathbf{T}}\left[ - \mathbf{x},t \right]\left<\mathbf{x}^{\prime}-\mathbf{x} \right> - - \underline{\mathbf{T}}\left[\mathbf{x}^{\prime},t \right]\left<\mathbf{x}-\mathbf{x}^{\prime} \right> \right\} - {d}V_{\mathbf{x}^\prime} + \mathbf{b}(\mathbf{x},t), + \int_{\mathcal{H}_{\mathbf{x}}} \left\{ + \underline{\mathbf{T}}\left[\mathbf{x},t + \right]\left<\mathbf{x}^{\prime}-\mathbf{x} \right> - + \underline{\mathbf{T}}\left[\mathbf{x}^{\prime},t + \right]\left<\mathbf{x}-\mathbf{x}^{\prime} \right> \right\} + {d}V_{\mathbf{x}^\prime} + \mathbf{b}(\mathbf{x},t), where :math:`\rho` represents the mass density, :math:`\underline{\mathbf{T}}` the force vector state, and @@ -147,7 +149,7 @@ where :math:`\rho` represents the mass density, radius :math:`\delta>0` centered at :math:`\mathbf{x}`. :math:`\delta` is called the *horizon*, and is analogous to the cutoff radius used in molecular dynamics. Conditions on :math:`\underline{\mathbf{T}}` for -which \eqref{eqn:NewtonII} satisfies the balance of linear and angular +which :math:numref:`periNewtonII` satisfies the balance of linear and angular momentum are given in :ref:`(Silling 2007) `. We consider only force vector states that can be written as @@ -161,6 +163,7 @@ with :math:`\underline{t}` a *scalar force state* and state*, defined by .. math:: + :label: periM \underline{\mathbf{M}}\left< \boldsymbol{\xi} \right> = \left\{ \begin{array}{cl} @@ -192,6 +195,7 @@ along with the horizon :math:`\delta`. The LPS model has a force scalar state .. math:: + :label: periLPSState \underline{t} = \frac{3K\theta}{m}\underline{\omega}\,\underline{x} + \alpha \underline{\omega}\,\underline{e}^{\rm d}, @@ -210,6 +214,7 @@ the reference position scalar state :math:`\underline{x}` so that defined as .. math:: + :label: periWeightedVolume m\left[ \mathbf{x} \right] = \int_{\mathcal{H}_\mathbf{x}} \underline{\omega} \left<\boldsymbol{\xi}\right> @@ -291,6 +296,7 @@ and the horizon :math:`\delta`. The PMB model is expressed using the scalar force state field .. math:: + :label: periPMBState \underline{t}\left[ \mathbf{x},t \right]\left< \boldsymbol{\xi} \right> = \frac{1}{2} f\left( \boldsymbol{\eta} ,\boldsymbol{\xi} \right), @@ -298,17 +304,21 @@ with :math:`f` a scalar-valued function. We assume that :math:`f` takes the form .. math:: + f = c s, where .. math:: + :label: peric + c = \frac{18K}{\pi \delta^4}, with :math:`K` the bulk modulus and :math:`\delta` the horizon, and :math:`s` the bond stretch, defined as .. math:: + :label: peristretch s(t,\mathbf{\eta},\mathbf{\xi}) = \frac{ \left\Vert {\mathbf{\eta}+\mathbf{\xi}} \right\Vert - \left\Vert {\mathbf{\xi}} \right\Vert }{\left\Vert {\mathbf{\xi}} \right\Vert}. @@ -320,9 +330,10 @@ appropriate for 3D models only. For more on the origins of the constant :math:`c`, see :ref:`(Silling 2005) `. For the derivation of :math:`c` for 1D and 2D models, see :ref:`(Emmrich) `. -Given \eqref{eqn:PMBState}, \eqref{eqn:NewtonII} reduces to +Given :math:numref:`periPMBState`, :math:numref:`periNewtonII` reduces to .. math:: + :label: periPMBEOM \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x},t) = \int_{\mathcal{H}_\mathbf{x}} \mathbf{f} \left( \boldsymbol{\eta},\boldsymbol{\xi} \right){d}V_{\boldsymbol{\xi}} + \mathbf{b}(\mathbf{x},t), @@ -352,6 +363,7 @@ Define :math:`\mu` to be the history-dependent scalar boolean function .. math:: + :label: perimu \mu(t,\mathbf{\eta},\mathbf{\xi}) = \left\{ \begin{array}{cl} @@ -366,6 +378,7 @@ where :math:`\mathbf{\eta}^\prime = \textbf{u}(\textbf{x}^{\prime critical stretch defined as .. math:: + :label: peris0 s_0(t,\mathbf{\eta},\mathbf{\xi}) = s_{00} - \alpha s_{\min}(t,\mathbf{\eta},\mathbf{\xi}), \qquad s_{\min}(t) = \min_{\mathbf{\xi}} s(t,\mathbf{\eta},\mathbf{\xi}), @@ -388,6 +401,7 @@ Following :ref:`(Silling) `, we can define the damage at a point :math:`\textbf{x}` as .. math:: + :label: peridamage \varphi(\textbf{x}, t) = 1 - \frac{\int_{\mathcal{H}_{\textbf{x}}} \mu(t,\mathbf{\eta},\mathbf{\xi}) dV_{\textbf{x}^\prime} }{ \int_{\mathcal{H}_{\textbf{x}}} dV_{\textbf{x}^\prime} }. @@ -395,7 +409,7 @@ point :math:`\textbf{x}` as Discrete Peridynamic Model and LAMMPS Implementation """""""""""""""""""""""""""""""""""""""""""""""""""" -In LAMMPS, instead of \eqref{eqn:NewtonII}, we model this equation of +In LAMMPS, instead of :math:numref:`periNewtonII`, we model this equation of motion: .. math:: @@ -409,7 +423,7 @@ where we explicitly track and store at each timestep the positions and not the displacements of the particles. We observe that :math:`\ddot{\textbf{y}}(\textbf{x}, t) = \ddot{\textbf{x}} + \ddot{\textbf{u}}(\textbf{x}, t) = \ddot{\textbf{u}}(\textbf{x}, t)`, so -that this is equivalent to \eqref{eqn:NewtonII}. +that this is equivalent to :math:numref:`periNewtonII`. Spatial Discretization ^^^^^^^^^^^^^^^^^^^^^^ @@ -422,12 +436,14 @@ denote the family of particles for which particle :math:`i` shares a bond in the reference configuration. That is, .. math:: + :label: periBondFamily \mathcal{F}_i = \{ p ~ | ~ \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert \leq \delta \}. -The discretized equation of motion replaces \eqref{eqn:NewtonII} with +The discretized equation of motion replaces :math:numref:`periNewtonII` with .. math:: + :label: peridiscreteNewtonII \rho \ddot{\textbf{y}}_i^n = \sum_{p \in \mathcal{F}_i} @@ -444,25 +460,29 @@ In the model discussed so far, particles interact only through their bond forces. A particle with no bonds becomes a free non-interacting particle. To account for contact forces, short-range forces are introduced :ref:`(Silling 2007) `. We add to the force in -\eqref{eqn:discreteNewtonII} the following force +:math:numref:`peridiscreteNewtonII` the following force .. math:: + :label: perifS \textbf{f}_S(\textbf{y}_p,\textbf{y}_i) = \min \left\{ 0, \frac{c_S}{\delta}(\left\Vert {\textbf{y}_p-\textbf{y}_i} \right\Vert - d_{pi}) \right\} \frac{\textbf{y}_p-\textbf{y}_i}{\left\Vert {\textbf{y}_p-\textbf{y}_i} \right\Vert}, where :math:`d_{pi}` is the short-range interaction distance between particles :math:`p` and :math:`i`, and :math:`c_S` is a multiple of the -constant :math:`c` from \eqref{eqn:c}. Note that the short-range force +constant :math:`c` from :math:numref:`peric`. Note that the short-range force is always repulsive, never attractive. In practice, we choose .. math:: + :label: pericS + c_S = 15 \frac{18K}{\pi \delta^4}. For the short-range interaction distance, we choose :ref:`(Silling 2007) ` .. math:: + :label: perid d_{pi} = \min \left\{ 0.9 \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert, 1.35 (r_p + r_i) \right\}, @@ -488,16 +508,16 @@ short-range family of particles Modification to the Particle Volume ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -The right-hand side of \eqref{eqn:discreteNewtonII} may be thought of as -a midpoint quadrature of \eqref{eqn:NewtonII}. To slightly improve the +The right-hand side of :math:numref:`peridiscreteNewtonII` may be thought of as +a midpoint quadrature of :math:numref:`periNewtonII`. To slightly improve the accuracy of this quadrature, we discuss a modification to the particle -volume used in \eqref{eqn:discreteNewtonII}. In a situation where two +volume used in :math:numref:`peridiscreteNewtonII`. In a situation where two particles share a bond with :math:`\left\Vert { \textbf{x}_p - \textbf{x}_i }\right\Vert = \delta`, for example, we suppose that only approximately half the volume of each particle is "seen" by the other :ref:`(Silling 2007) `. When computing the force of each particle on the other we use :math:`V_p / 2` rather than :math:`V_p` in -\eqref{eqn:discreteNewtonII}. As such, we introduce a nodal volume +:math:numref:`peridiscreteNewtonII`. As such, we introduce a nodal volume scaling function for all bonded particles where :math:`\delta - r_i \leq \left\Vert { \textbf{x}_p - \textbf{x}_i } \right\Vert \leq \delta` (see the Figure below). @@ -555,12 +575,12 @@ Breaking Bonds During the course of simulation, it may be necessary to break bonds, as described in the :ref:`Damage section `. Bonds are recorded as broken in a simulation by removing them from the bond family -:math:`\mathcal{F}_i` (see \eqref{eqn:BondFamily}). +:math:`\mathcal{F}_i` (see :math:numref:`periBondFamily`). A naive implementation would have us first loop over all bonds and -compute :math:`s_{min}` in \eqref{eqn:s0}, then loop over all bonds +compute :math:`s_{min}` in :math:numref:`peris0`, then loop over all bonds again and break bonds with a stretch :math:`s > s0` as in -\eqref{eqn:mu}, and finally loop over all particles and compute forces +:math:numref:`perimu`, and finally loop over all particles and compute forces for the next step of :ref:`Algorithm 1 `. For reasons of computational efficiency, we will utilize the values of :math:`s_0` from the *previous* timestep when deciding to break a bond. @@ -602,7 +622,7 @@ routines in :ref:`Algorithm 3 ` and :ref:`Algorithm 4 | **for all** particles :math:`j \in \mathcal{F}^S_i` (the short-range family of nodes for particle :math:`i`) **do** | :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert` | :math:`dr = \min \{ 0, r - d \}` {*Short-range forces are only repulsive, never attractive*} - | :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in \eqref{eqn:cS}*} + | :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in :math:numref:`pericS`*} | :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}` | **end for** | **end for** @@ -675,7 +695,7 @@ A sketch of the PMB model implementation in the PERI package appears in | **for all** particles :math:`j \in \mathcal{F}^S_i` (the short-range family of nodes for particle :math:`i`) **do** | :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert` | :math:`dr = \min \{ 0, r - d \}` {*Short-range forces are only repulsive, never attractive*} - | :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in \eqref{eqn:cS}*} + | :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in :math:numref:`pericS`*} | :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}` | **end for** | **end for** @@ -684,7 +704,7 @@ A sketch of the PMB model implementation in the PERI package appears in | **for all** particles :math:`j` sharing an unbroken bond with particle :math:`i` **do** | :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert` | :math:`dr = r - \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert` - | :math:`k = \frac{c}{\left\Vert {\textbf{x}_i - \textbf{x}_j} \right\Vert} \nu(\textbf{x}_i - \textbf{x}_j) V_j dr` {:math:`c` *defined in \eqref{eqn:c}*} + | :math:`k = \frac{c}{\left\Vert {\textbf{x}_i - \textbf{x}_j} \right\Vert} \nu(\textbf{x}_i - \textbf{x}_j) V_j dr` {:math:`c` *defined in :math:numref:`peric`*} | :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}` | **if** :math:`(dr / \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert) > \min(s_0(i), s_0(j))` **then** | Break :math:`i`'s bond with :math:`j` {:math:`j`\ *'s bond with* :math:`i` *will be broken when this loop iterates on* :math:`j`} @@ -699,7 +719,7 @@ A sketch of the PMB model implementation in the PERI package appears in Damage ^^^^^^ -The damage associated with every particle (see \eqref{eqn:damage}) can +The damage associated with every particle (see :math:numref:`peridamage`) can optionally be computed and output with a LAMMPS data dump. To do this, your input script must contain the command :doc:`compute damage/atom ` This enables a LAMMPS per-atom compute to @@ -880,7 +900,7 @@ three times the lattice constant.) The target is a cylinder of diameter lattice contains 103,110 particles. Each particle :math:`i` has volume fraction :math:`V_i = 1.25 \times 10^{-10} \textrm{m}^3`. -The spring constant in the PMB material model is (see \eqref{eqn:c}) +The spring constant in the PMB material model is (see :math:numref:`peric`) .. math:: c = \frac{18k}{\pi \delta^4} = \frac{18 (14.9 \times 10^9)}{ \pi (1.5 \times 10^{-3})^4} \approx 1.6863 \times 10^{22}. @@ -927,7 +947,7 @@ the distance from the atom to the center of the indenter, and :math:`R` is the radius of the projectile. The force is repulsive and :math:`F(r) = 0` for :math:`r > R`. For our problem, the projectile radius :math:`R = 0.05` m, and we have chosen :math:`k_s = 1.0 \times 10^{17}` (compare -with \eqref{eqn:c} above). +with :math:numref:`peric` above). Writing the LAMMPS Input File ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ diff --git a/doc/utils/sphinx-config/_static/css/lammps.css b/doc/utils/sphinx-config/_static/css/lammps.css index cbf08c3da1..2a5b425535 100644 --- a/doc/utils/sphinx-config/_static/css/lammps.css +++ b/doc/utils/sphinx-config/_static/css/lammps.css @@ -1,3 +1,11 @@ +.math { + text-align: left; +} + +.eqno { + float: right; +} + .wy-nav-content { max-width: 100% !important; } From 09948f11bed1f9f2657319dce212f0eb0c1e8f31 Mon Sep 17 00:00:00 2001 From: Axel Kohlmeyer Date: Thu, 29 Sep 2022 13:54:09 -0400 Subject: [PATCH 4/4] make equation references explicit since .. math:: + :label: does not work with latex --- doc/src/Howto_peri.rst | 119 +++++++++++++++++++++-------------------- 1 file changed, 61 insertions(+), 58 deletions(-) diff --git a/doc/src/Howto_peri.rst b/doc/src/Howto_peri.rst index ff303f16aa..4c78d04833 100644 --- a/doc/src/Howto_peri.rst +++ b/doc/src/Howto_peri.rst @@ -129,8 +129,9 @@ collectively on all points interacting with that point. Using the notation of :ref:`(Silling 2007) `, we write the peridynamic equation of motion as +.. _periNewtonII: + .. math:: - :label: periNewtonII \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x},t) = \int_{\mathcal{H}_{\mathbf{x}}} \left\{ @@ -138,7 +139,7 @@ peridynamic equation of motion as \right]\left<\mathbf{x}^{\prime}-\mathbf{x} \right> - \underline{\mathbf{T}}\left[\mathbf{x}^{\prime},t \right]\left<\mathbf{x}-\mathbf{x}^{\prime} \right> \right\} - {d}V_{\mathbf{x}^\prime} + \mathbf{b}(\mathbf{x},t), + {d}V_{\mathbf{x}^\prime} + \mathbf{b}(\mathbf{x},t), \qquad\qquad\textrm{(1)} where :math:`\rho` represents the mass density, :math:`\underline{\mathbf{T}}` the force vector state, and @@ -149,7 +150,7 @@ where :math:`\rho` represents the mass density, radius :math:`\delta>0` centered at :math:`\mathbf{x}`. :math:`\delta` is called the *horizon*, and is analogous to the cutoff radius used in molecular dynamics. Conditions on :math:`\underline{\mathbf{T}}` for -which :math:numref:`periNewtonII` satisfies the balance of linear and angular +which :ref:`(1) ` satisfies the balance of linear and angular momentum are given in :ref:`(Silling 2007) `. We consider only force vector states that can be written as @@ -163,7 +164,6 @@ with :math:`\underline{t}` a *scalar force state* and state*, defined by .. math:: - :label: periM \underline{\mathbf{M}}\left< \boldsymbol{\xi} \right> = \left\{ \begin{array}{cl} @@ -172,7 +172,7 @@ state*, defined by } & \left\Vert \boldsymbol{\xi} + \boldsymbol{\eta} \right\Vert \neq 0 \\ \boldsymbol{0} & \textrm{otherwise} \end{array} - \right. . + \right. . \qquad\qquad\textrm{(2)} Such force states correspond to so-called *ordinary* materials :ref:`(Silling 2007) `. These are the materials for which @@ -195,10 +195,9 @@ along with the horizon :math:`\delta`. The LPS model has a force scalar state .. math:: - :label: periLPSState \underline{t} = \frac{3K\theta}{m}\underline{\omega}\,\underline{x} + - \alpha \underline{\omega}\,\underline{e}^{\rm d}, + \alpha \underline{\omega}\,\underline{e}^{\rm d}, \qquad\qquad\textrm{(3)} with :math:`K` the bulk modulus and :math:`\alpha` related to the shear modulus :math:`G` as @@ -214,12 +213,11 @@ the reference position scalar state :math:`\underline{x}` so that defined as .. math:: - :label: periWeightedVolume m\left[ \mathbf{x} \right] = \int_{\mathcal{H}_\mathbf{x}} \underline{\omega} \left<\boldsymbol{\xi}\right> \underline{x}\left<\boldsymbol{\xi} \right> - \underline{x}\left<\boldsymbol{\xi} \right>{d}V_{\boldsymbol{\xi} }. + \underline{x}\left<\boldsymbol{\xi} \right>{d}V_{\boldsymbol{\xi} }. \qquad\qquad\textrm{(4)} Let @@ -295,10 +293,11 @@ and the horizon :math:`\delta`. The PMB model is expressed using the scalar force state field -.. math:: - :label: periPMBState +.. _periPMBState: - \underline{t}\left[ \mathbf{x},t \right]\left< \boldsymbol{\xi} \right> = \frac{1}{2} f\left( \boldsymbol{\eta} ,\boldsymbol{\xi} \right), +.. math:: + + \underline{t}\left[ \mathbf{x},t \right]\left< \boldsymbol{\xi} \right> = \frac{1}{2} f\left( \boldsymbol{\eta} ,\boldsymbol{\xi} \right), \qquad\qquad\textrm{(5)} with :math:`f` a scalar-valued function. We assume that :math:`f` takes the form @@ -309,16 +308,16 @@ the form where -.. math:: - :label: peric +.. _peric: - c = \frac{18K}{\pi \delta^4}, +.. math:: + + c = \frac{18K}{\pi \delta^4}, \qquad\qquad\textrm{(6)} with :math:`K` the bulk modulus and :math:`\delta` the horizon, and :math:`s` the bond stretch, defined as .. math:: - :label: peristretch s(t,\mathbf{\eta},\mathbf{\xi}) = \frac{ \left\Vert {\mathbf{\eta}+\mathbf{\xi}} \right\Vert - \left\Vert {\mathbf{\xi}} \right\Vert }{\left\Vert {\mathbf{\xi}} \right\Vert}. @@ -330,12 +329,11 @@ appropriate for 3D models only. For more on the origins of the constant :math:`c`, see :ref:`(Silling 2005) `. For the derivation of :math:`c` for 1D and 2D models, see :ref:`(Emmrich) `. -Given :math:numref:`periPMBState`, :math:numref:`periNewtonII` reduces to +Given :ref:`(5) `, :ref:`(1) ` reduces to .. math:: - :label: periPMBEOM - \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x},t) = \int_{\mathcal{H}_\mathbf{x}} \mathbf{f} \left( \boldsymbol{\eta},\boldsymbol{\xi} \right){d}V_{\boldsymbol{\xi}} + \mathbf{b}(\mathbf{x},t), + \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x},t) = \int_{\mathcal{H}_\mathbf{x}} \mathbf{f} \left( \boldsymbol{\eta},\boldsymbol{\xi} \right){d}V_{\boldsymbol{\xi}} + \mathbf{b}(\mathbf{x},t), \qquad\qquad\textrm{(7)} with @@ -362,14 +360,15 @@ the *critical stretch* criterion. Define :math:`\mu` to be the history-dependent scalar boolean function +.. _perimu: + .. math:: - :label: perimu \mu(t,\mathbf{\eta},\mathbf{\xi}) = \left\{ \begin{array}{cl} 1 & \mbox{if $s(t^\prime,\mathbf{\eta},\mathbf{\xi})< \min \left(s_0(t^\prime,\mathbf{\eta},\mathbf{\xi}) , s_0(t^\prime,\mathbf{\eta}^\prime,\mathbf{\xi}^\prime) \right)$ for all $0 \leq t^\prime \leq t$} \\ 0 & \mbox{otherwise} - \end{array}\right\}. + \end{array}\right\}. \qquad\qquad\textrm{(8)} where :math:`\mathbf{\eta}^\prime = \textbf{u}(\textbf{x}^{\prime \prime},t) - \textbf{u}(\textbf{x}^\prime,t)` and @@ -377,10 +376,11 @@ where :math:`\mathbf{\eta}^\prime = \textbf{u}(\textbf{x}^{\prime \textbf{x}^\prime`. Here, :math:`s_0(t,\mathbf{\eta},\mathbf{\xi})` is a critical stretch defined as -.. math:: - :label: peris0 +.. _peris0: - s_0(t,\mathbf{\eta},\mathbf{\xi}) = s_{00} - \alpha s_{\min}(t,\mathbf{\eta},\mathbf{\xi}), \qquad s_{\min}(t) = \min_{\mathbf{\xi}} s(t,\mathbf{\eta},\mathbf{\xi}), +.. math:: + + s_0(t,\mathbf{\eta},\mathbf{\xi}) = s_{00} - \alpha s_{\min}(t,\mathbf{\eta},\mathbf{\xi}), \qquad s_{\min}(t) = \min_{\mathbf{\xi}} s(t,\mathbf{\eta},\mathbf{\xi}), \qquad\qquad\textrm{(9)} where :math:`s_{00}` and :math:`\alpha` are material-dependent constants. The history function :math:`\mu` breaks bonds when the @@ -400,16 +400,17 @@ should be broken. Following :ref:`(Silling) `, we can define the damage at a point :math:`\textbf{x}` as +.. _peridamageeq: + .. math:: - :label: peridamage \varphi(\textbf{x}, t) = 1 - \frac{\int_{\mathcal{H}_{\textbf{x}}} \mu(t,\mathbf{\eta},\mathbf{\xi}) dV_{\textbf{x}^\prime} - }{ \int_{\mathcal{H}_{\textbf{x}}} dV_{\textbf{x}^\prime} }. + }{ \int_{\mathcal{H}_{\textbf{x}}} dV_{\textbf{x}^\prime} }. \qquad\qquad\textrm{(10)} Discrete Peridynamic Model and LAMMPS Implementation """""""""""""""""""""""""""""""""""""""""""""""""""" -In LAMMPS, instead of :math:numref:`periNewtonII`, we model this equation of +In LAMMPS, instead of :ref:`(1) `, we model this equation of motion: .. math:: @@ -423,7 +424,7 @@ where we explicitly track and store at each timestep the positions and not the displacements of the particles. We observe that :math:`\ddot{\textbf{y}}(\textbf{x}, t) = \ddot{\textbf{x}} + \ddot{\textbf{u}}(\textbf{x}, t) = \ddot{\textbf{u}}(\textbf{x}, t)`, so -that this is equivalent to :math:numref:`periNewtonII`. +that this is equivalent to :ref:`(1) `. Spatial Discretization ^^^^^^^^^^^^^^^^^^^^^^ @@ -435,21 +436,23 @@ where each particle :math:`i` is associated with some volume fraction denote the family of particles for which particle :math:`i` shares a bond in the reference configuration. That is, -.. math:: - :label: periBondFamily - - \mathcal{F}_i = \{ p ~ | ~ \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert \leq \delta \}. - -The discretized equation of motion replaces :math:numref:`periNewtonII` with +.. _periBondFamily: + +.. math:: + + \mathcal{F}_i = \{ p ~ | ~ \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert \leq \delta \}. \qquad\qquad\textrm{(11)} + +The discretized equation of motion replaces :ref:`(1) ` with + +.. _peridiscreteNewtonII: .. math:: - :label: peridiscreteNewtonII \rho \ddot{\textbf{y}}_i^n = \sum_{p \in \mathcal{F}_i} \left\{ \underline{\mathbf{T}}\left[ \textbf{x}_i,t \right]\left<\textbf{x}_p^{\prime}-\textbf{x}_i \right> - \underline{\mathbf{T}}\left[\textbf{x}_p,t \right]\left<\textbf{x}_i-\textbf{x}_p \right> \right\} - V_{p} + \textbf{b}_i^n, + V_{p} + \textbf{b}_i^n, \qquad\qquad\textrm{(12)} where :math:`n` is the timestep number and subscripts denote the particle number. @@ -460,31 +463,30 @@ In the model discussed so far, particles interact only through their bond forces. A particle with no bonds becomes a free non-interacting particle. To account for contact forces, short-range forces are introduced :ref:`(Silling 2007) `. We add to the force in -:math:numref:`peridiscreteNewtonII` the following force +:ref:`(12) ` the following force .. math:: - :label: perifS \textbf{f}_S(\textbf{y}_p,\textbf{y}_i) = \min \left\{ 0, \frac{c_S}{\delta}(\left\Vert {\textbf{y}_p-\textbf{y}_i} \right\Vert - d_{pi}) \right\} - \frac{\textbf{y}_p-\textbf{y}_i}{\left\Vert {\textbf{y}_p-\textbf{y}_i} \right\Vert}, + \frac{\textbf{y}_p-\textbf{y}_i}{\left\Vert {\textbf{y}_p-\textbf{y}_i} \right\Vert}, \qquad\qquad\textrm{(13)} where :math:`d_{pi}` is the short-range interaction distance between particles :math:`p` and :math:`i`, and :math:`c_S` is a multiple of the -constant :math:`c` from :math:numref:`peric`. Note that the short-range force +constant :math:`c` from :ref:`(6) `. Note that the short-range force is always repulsive, never attractive. In practice, we choose -.. math:: - :label: pericS +.. _pericS: - c_S = 15 \frac{18K}{\pi \delta^4}. +.. math:: + + c_S = 15 \frac{18K}{\pi \delta^4}. \qquad\qquad\textrm{(14)} For the short-range interaction distance, we choose :ref:`(Silling 2007) ` .. math:: - :label: perid - d_{pi} = \min \left\{ 0.9 \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert, 1.35 (r_p + r_i) \right\}, + d_{pi} = \min \left\{ 0.9 \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert, 1.35 (r_p + r_i) \right\}, \qquad\qquad\textrm{(15)} where :math:`r_i` is called the *node radius* of particle :math:`i`. Given a discrete lattice, we choose :math:`r_i` to be half @@ -508,16 +510,16 @@ short-range family of particles Modification to the Particle Volume ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -The right-hand side of :math:numref:`peridiscreteNewtonII` may be thought of as -a midpoint quadrature of :math:numref:`periNewtonII`. To slightly improve the +The right-hand side of :ref:`(12) ` may be thought of as +a midpoint quadrature of :ref:`(1) `. To slightly improve the accuracy of this quadrature, we discuss a modification to the particle -volume used in :math:numref:`peridiscreteNewtonII`. In a situation where two +volume used in :ref:`(12) `. In a situation where two particles share a bond with :math:`\left\Vert { \textbf{x}_p - \textbf{x}_i }\right\Vert = \delta`, for example, we suppose that only approximately half the volume of each particle is "seen" by the other :ref:`(Silling 2007) `. When computing the force of each particle on the other we use :math:`V_p / 2` rather than :math:`V_p` in -:math:numref:`peridiscreteNewtonII`. As such, we introduce a nodal volume +:ref:`(12) `. As such, we introduce a nodal volume scaling function for all bonded particles where :math:`\delta - r_i \leq \left\Vert { \textbf{x}_p - \textbf{x}_i } \right\Vert \leq \delta` (see the Figure below). @@ -575,12 +577,12 @@ Breaking Bonds During the course of simulation, it may be necessary to break bonds, as described in the :ref:`Damage section `. Bonds are recorded as broken in a simulation by removing them from the bond family -:math:`\mathcal{F}_i` (see :math:numref:`periBondFamily`). +:math:`\mathcal{F}_i` (see :ref:`(11) `). A naive implementation would have us first loop over all bonds and -compute :math:`s_{min}` in :math:numref:`peris0`, then loop over all bonds +compute :math:`s_{min}` in :ref:`(9) `, then loop over all bonds again and break bonds with a stretch :math:`s > s0` as in -:math:numref:`perimu`, and finally loop over all particles and compute forces +:ref:`(8) `, and finally loop over all particles and compute forces for the next step of :ref:`Algorithm 1 `. For reasons of computational efficiency, we will utilize the values of :math:`s_0` from the *previous* timestep when deciding to break a bond. @@ -622,7 +624,7 @@ routines in :ref:`Algorithm 3 ` and :ref:`Algorithm 4 | **for all** particles :math:`j \in \mathcal{F}^S_i` (the short-range family of nodes for particle :math:`i`) **do** | :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert` | :math:`dr = \min \{ 0, r - d \}` {*Short-range forces are only repulsive, never attractive*} - | :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in :math:numref:`pericS`*} + | :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in :ref:`(14) `*} | :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}` | **end for** | **end for** @@ -695,7 +697,7 @@ A sketch of the PMB model implementation in the PERI package appears in | **for all** particles :math:`j \in \mathcal{F}^S_i` (the short-range family of nodes for particle :math:`i`) **do** | :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert` | :math:`dr = \min \{ 0, r - d \}` {*Short-range forces are only repulsive, never attractive*} - | :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in :math:numref:`pericS`*} + | :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in :ref:`(14) `*} | :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}` | **end for** | **end for** @@ -704,7 +706,7 @@ A sketch of the PMB model implementation in the PERI package appears in | **for all** particles :math:`j` sharing an unbroken bond with particle :math:`i` **do** | :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert` | :math:`dr = r - \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert` - | :math:`k = \frac{c}{\left\Vert {\textbf{x}_i - \textbf{x}_j} \right\Vert} \nu(\textbf{x}_i - \textbf{x}_j) V_j dr` {:math:`c` *defined in :math:numref:`peric`*} + | :math:`k = \frac{c}{\left\Vert {\textbf{x}_i - \textbf{x}_j} \right\Vert} \nu(\textbf{x}_i - \textbf{x}_j) V_j dr` {:math:`c` *defined in :ref:`(6) `*} | :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}` | **if** :math:`(dr / \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert) > \min(s_0(i), s_0(j))` **then** | Break :math:`i`'s bond with :math:`j` {:math:`j`\ *'s bond with* :math:`i` *will be broken when this loop iterates on* :math:`j`} @@ -719,7 +721,7 @@ A sketch of the PMB model implementation in the PERI package appears in Damage ^^^^^^ -The damage associated with every particle (see :math:numref:`peridamage`) can +The damage associated with every particle (see :ref:`(10) `) can optionally be computed and output with a LAMMPS data dump. To do this, your input script must contain the command :doc:`compute damage/atom ` This enables a LAMMPS per-atom compute to @@ -900,9 +902,10 @@ three times the lattice constant.) The target is a cylinder of diameter lattice contains 103,110 particles. Each particle :math:`i` has volume fraction :math:`V_i = 1.25 \times 10^{-10} \textrm{m}^3`. -The spring constant in the PMB material model is (see :math:numref:`peric`) +The spring constant in the PMB material model is (see :ref:`(6) `) .. math:: + c = \frac{18k}{\pi \delta^4} = \frac{18 (14.9 \times 10^9)}{ \pi (1.5 \times 10^{-3})^4} \approx 1.6863 \times 10^{22}. The CFL analysis from :ref:`(Silling2005) ` shows that a @@ -947,7 +950,7 @@ the distance from the atom to the center of the indenter, and :math:`R` is the radius of the projectile. The force is repulsive and :math:`F(r) = 0` for :math:`r > R`. For our problem, the projectile radius :math:`R = 0.05` m, and we have chosen :math:`k_s = 1.0 \times 10^{17}` (compare -with :math:numref:`peric` above). +with :ref:`(6) ` above). Writing the LAMMPS Input File ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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