Deborah Sulsky

A particle method for history-dependent materials

Abstract A broad class of engineering problems including penetration, impact and large rotations of solid bodies causes severe numerical problems. For these problems, the constitutive equations are history dependent so material points must be followed; this is difficult to implement in a Eulerian scheme. On the other hand, purely Lagrangian methods typically result in severe mesh distortion and the consequence is ill conditioning of the element stiffness matrix leading to mesh lockup or entanglement. Remeshing prevents the lockup and tangling but then interpolation must be performed for history dependent variables, a process which can introduce errors. Proposed here is an extension of the particle-in-cell method in which particles are interpreted to be material points that are followed through the complete loading process. A fixed Eulerian grid provides the means for determining a spatial gradient. Because the grid can also be interpreted as an updated Lagrangian frame, the usual convection term in the acceleration associated with Eulerian formulations does not appear. With the use of maps between material points and the grid, the advantages of both Eulerian and Lagrangian schemes are utilized so that mesh tangling is avoided while material variables are tracked through the complete deformation history. Example solutions in two dimensions are given to illustrate the robustness of the proposed convection algorithm and to show that typical elastic behavior can be reproduced. Also, it is shown that impact with no slip is handled without any special algorithm for bodies governed by elasticity and strain hardening plasticity.

The material-point method for granular materials

Abstract A model for granular materials is presented that describes both the internal deformation of each granule and the interactions between grains. The model, which is based on the FLIP-material point, particle-in-cell method, solves continuum constutitive models for each grain. Interactions between grains are calculated with a contact algorithm that forbids interpenetration, but allows separation and sliding and rolling with friction. The particle-in-cell method eliminates the need for a separate contact detection step. The use of a common rest frame in the contact model yields a linear scaling of the computational cost with the number of grains. The properties of the model are illustrated by numerical solutions of sliding and rolling contacts, and for granular materials by a shear calculation. The results of numerical calculations demonstrate that contacts are modeled accurately for smooth granules whose shape is resolved by the computations mesh.

Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems

Abstract The material point method is an evolution of particle-in-cell methods which utilize two meshes, one a material or Lagrangian mesh defined over material of the body under consideration, and the second a spatial or Eulerian mesh defined over the computational domain. Although meshes are used, they have none of the negative aspects normally associated with conventional Eulerian or Lagrangian approaches. The advantages of both the Eulerian and Lagrangian methods are achieved by using the appropriate frame for each aspect of the computation, with a mapping between the two meshes that is performed at each step in the loading process. The numerical dissipation normally displayed by an Eulerian method because of advection is avoided by using a Lagrangian step; the mesh distortion associated with the Lagrangian method is prevented by mapping to a user-controlled mesh. Furthermore, explicit material points can be tracked through the process of deformation, thereby alleviating the need to map history variables. As a consequence, problems which have caused severe numerical difficulties with conventional methods are handled fairly routinely. Examples of such problems are the upsetting of billets and the Taylor problem of cylinders impacting a rigid wall. Numerical solutions to these problems are obtained with the material point method and where possible comparisons with experimental data and existing numerical solutions are presented.

Fluid–membrane interaction based on the material point method

The material point method (MPM) uses unconnected, Lagrangian, material points to discretize solids, fluids or membranes. All variables in the solution of the continuum equations are associated with these points; so, for example, they carry mass, velocity, stress and strain. A background Eulerian mesh is used to solve the momentum equation. Data mapped from the material points are used to initialize variables on the background mesh. In the case of multiple materials, the stress from each material contributes to forces at nearby mesh points, so the solution of the momentum equation includes all materials. The mesh solution then updates the material point values. This simple algorithm treats all materials in a uniform way, avoids complicated mesh construction and automatically applies a noslip contact algorithm at no additional cost. Several examples are used to demonstrate the method, including simulation of a pressurized membrane and the impact of a probe with a pre-inflated airbag. Copyright

A numerical method for suspension flow

Abstract Peskins ( J. Comput. Phys. 25 , 220, 1977) immersed boundary technique is modified to give a new numerical method for studying a fluid with suspended elastic particles. As before, the presence of the suspended particles is transmitted to the fluid through a force density term in the fluid equations. As a result, one set of equations holds in the entire computational domain, eliminating the need to apply boundary conditions on the surface of suspended objects. The new method computes the force density by discretizing the stress-strain constitutive equations for an elastic solid on a grid, using data provided by clusters of Lagrangian points. This approach clearly specifies the material properties of the suspended objects. A simple data structure for the Lagrangian points makes it easy to model suspended solids with arbitrary shape and size. The method is validated by comparing numerical results for elastic vibrations and particle settling in viscous fluids, with theory and analysis. The capability of the method to do a wide range of problems is illustrated by qualitative results for lubrication and cavity flow problems.

Mass matrix formulation of the FLIP particle-in-cell method

A refinement of FLIP [Brackbill and Ruppel, J. Comput. Phys. 65, 314 (1986)] is described which uses a mass matrix formulation to achieve greater accuracy and less numerical diffusion over the previous version. Without the refinement, there is a significant dissipation of energy in modeling elastic vibrations of a solid. Moreover, in modeling an initial flow discontinuity there are sub-grid-scale oscillations in the particle velocity field in the neighborhood of the discontinuity. These difficulties are eliminated using the mass matrix. In addition, the mass matrix formulation conserves kinetic energy, linear and angular momentum, and is Galilean invariant.

Using the material‐point method to model sea ice dynamics

[1] The material-point method (MPM) is a numerical method for continuum mechanics that combines the best aspects of Lagrangian and Eulerian discretizations. The material points provide a Lagrangian description of the ice that models convection naturally. Thus properties such as ice thickness and compactness are computed in a Lagrangian frame and do not suffer from errors associated with Eulerian advection schemes, such as artificial diffusion, dispersion, or oscillations near discontinuities. This desirable property is illustrated by solving transport of ice in uniform, rotational and convergent velocity fields. Moreover, the ice geometry is represented by unconnected material points rather than a grid. This representation facilitates modeling the large deformations observed in the Arctic, as well as localized deformation along leads, and admits a sharp representation of the ice edge. MPM also easily allows the use of any ice constitutive model. The versatility of MPM is demonstrated by using two constitutive models for simulations of wind-driven ice. The first model is a standard viscous-plastic model with two thickness categories. The MPM solution to the viscous-plastic model agrees with previously published results using finite elements. The second model is a new elastic-decohesive model that explicitly represents leads. The model includes a mechanism to initiate leads, and to predict their orientation and width. The elastic-decohesion model can provide similar overall deformation as the viscous-plastic model; however, explicit regions of opening and shear are predicted. Furthermore, the efficiency of MPM with the elastic-decohesive model is competitive with the current best methods for sea ice dynamics.

The material point method for simulation of thin membranes

The material-point method (MPM) is extended to handle membranes, which are discretized by a collection of unconnected material points placed along each membrane surface. These points provide a Lagrangian description of the membrane. To solve for the membrane motion, data carried by the material points are transferred to a background mesh where equations of motion are discretized and solved. Then the solution on the background mesh is used to update the membrane material points. This process of combining Lagrangian and Eulerian features is standard in MPM; the modification for membranes involves merely an implementation of the constitutive equation in a local, normal-tangential coordinate system. It is shown that this procedure does, in fact, provide adequate resolution of membranes with thicknesses that can vary substantially from that of the background mesh spacing. A general formulation is given, but the implementation is in a two-dimensional code that provides a proof-of-principle. Numerical examples including a spring, pendulum and a string with initial slack are used to illustrate the method. The string with slack uses an additional modification of the membrane constitutive equation that allows wrinkles to be modeled at low computational cost. Presented also are examples of two disks impacting, pinching a membrane and rebounding, a difficult problem for standard finite element codes. These simulations require a relaxation of the automatic no-slip contact algorithm in MPM. The addition of the capability to model membranes and the new contact algorithm provide a significant improvement over existing methods for handling an important class of problems. Copyright

Arctic Ice Dynamics Joint Experiment (AIDJEX) assumptions revisited and found inadequate

[1] This paper revisits the Arctic Ice Dynamics Joint Experiment (AIDJEX) assumptions about pack ice behavior with an eye to modeling sea ice dynamics. The AIDJEX assumptions were that (1) enough leads were present in a 100 km by 100 km region to make the ice isotropic on that scale; (2) the ice had no tensile strength; and (3) the ice behavior could be approximated by an isotropic yield surface. These assumptions were made during the development of the AIDJEX model in the 1970s, and are now found inadequate. The assumptions were made in part because of insufficient large-scale (10 km) deformation and stress data, and in part because of computer capability limitations. Upon reviewing deformation and stress data, it is clear that a model including deformation on discontinuities and an anisotropic failure surface with tension would better describe the behavior of pack ice. A model based on these assumptions is needed to represent the deformation and stress in pack ice on scales from 10 to 100 km, and would need to explicitly resolve discontinuities. Such a model would require a different class of metrics to validate discontinuities against observations.

Interfacing continuum and molecular dynamics: An application to lipid bilayers

A new methodology is presented for interfacing atomistic molecular dynamics simulations with continuum dynamics, and the approach is then applied to a model lipid bilayer system. The technique relies on a closed feedback loop in which atomistic level simulations are coupled with continuum level modeling. This approach allows for the examination of the trans-temporal and trans-spatial phenomena that occur in many biological systems, and nonequilibrium molecular dynamics are used to calculate the relevant transport coefficients that are required at the continuum level. It is found that for the membrane system there is significant information transfer across disparate spatial and temporal regimes, resulting in significant nonlinear behavior.