Integrating Peridynamics with Material Point Method for Elastoplastic Material Modeling

© Springer Nature Switzerland AG 2019. We present a novel integral-based Material Point Method (MPM) using state based peridynamics structure for modeling elastoplastic material and fracture animation. Previous partial derivative based MPM studies face challenges of underlying instability issues of particle distribution and the complexity of modeling discontinuities. To alleviate these problems, we integrate the strain metric in the basic elastic constitutive model by using material point truss structure, which outweighs differential-based methods in both accuracy and stability. To model plasticity, we incorporate our constitutive model with deviatoric flow theory and a simple yield function. It is straightforward to handle the problem of cracking in our hybrid framework. Our method adopts two time integration ways to update crack interface and fracture inner parts, which overcome the unnecessary grid duplication. Our work can create a wide range of material phenomenon including elasticity, plasticity, and fracture. Our framework provides an attractive method for producing elastoplastic materials and fracture with visual realism and high stability

A Material Point Method for Elastoplasticity with Ductile Fracture and Frictional Contact

Simulating physical materials with dynamic movements to photo-realistic resolution has always been one of the most crucial and challenging topics in Computer Graphics. This dissertation considers large-strain elastoplasticity theory applied to the low-to-medium stiffness regime, with topological changes and codimensional objects incorporated. We introduce improvements to the Material Point Method (MPM) for two particular objectives, simulating fracturing ductile materials and incorporation of MPM and Lagrangian Finite Element Method (FEM).Our first contribution, simulating ductile fracture, utilizes traditional particle-based MPM [SSC13, SCS94] as well as the Lagrangian energy formulation of [JSS15] which uses a tetrahedron mesh, rather than particle-based estimation of the deformation gradient and potential energy. We model failure and fracture via elastoplasticity with damage. The material is elastic until its deformation exceeds a Rankine or von Mises yield condition. At that point, we use a softening model that shrinks the yield surface until it reaches the damage thresh- old. Once damaged, the material Lam ́e coefficients are modified to represent failed material. This approach to simulating ductile fracture with MPM is successful, as MPM naturally captures the topological changes coming from the fracture. However, rendering the crack surfaces can be challenging. We design a novel visualization technique dedicated to rendering the material’s boundary and its intersection with the evolving crack surfaces. Our approach uses a simple and efficient element splitting strategy for tetrahedron meshes to create crack surfaces. It employs an extrapolation technique based on the MPM simulation. For traditional particle-based MPM, we use an initial Delaunay tetrahedralization to connect randomly sampled MPM particles. Our visualization technique is a post-process and can run after the MPM simulation for efficiency. We demonstrate our method with several challenging simulations of ductile failure with considerable and persistent self-contact and applications with thermomechanical models for baking and cooking.Our second contribution, hybrid MPM–Lagrangian-FEM, aims to simulate elastic objects like hair, rubber, and soft tissues. It utilizes a Lagrangian mesh for internal force computation and a Eulerian grid for self-collision, as well as coupling with external materials. While recent MPM techniques allow for natural simulation of hyperelastic materials represented with Lagrangian meshes, they utilize an updated Lagrangian discretization and use the Eulerian grid degrees of freedom to take variations of the potential energy. It often coarsens the degrees of freedom of the Lagrangian mesh and can lead to artifacts. We develop a hybrid approach that retains Lagrangian degrees of freedom while still allowing for natural coupling with other materials simulated with traditional MPM, e.g., sand, snow, etc. Furthermore, while recent MPM advances allow for resolution of frictional contact with codimensional simulation of hyperelasticity, they do not generalize to the case of volumetric materials. We show that our hybrid approach resolves these issues. We demonstrate the efficacy of our technique with examples that involve elastic soft tissues coupled with kinematic skeletons, extreme deformation, and coupling with various elastoplastic materials. Our approach also naturally allows for two-way rigid body coupling

Variational Bonded Discrete Element Method with Manifold Optimization

This paper proposes a novel approach that combines variational integration with the bonded discrete element method (BDEM) to achieve faster and more accurate fracture simulations. The approach leverages the efficiency of implicit integration and the accuracy of BDEM in modeling fracture phenomena. We introduce a variational integrator and a manifold optimization approach utilizing a nullspace operator to speed up the solving of quaternion-constrained systems. Additionally, the paper presents an element packing and surface reconstruction method specifically designed for bonded discrete element methods. Results from the experiments prove that the proposed method offers 2.8 to 12 times faster state-of-the-art methods

Integral-based Material Point Method and Peridynamics Model For Animating Elastoplastic Material

This paper exploits the use of Material Point Method (MPM) for graphical animation of elastoplastic materials and fracture. Previous partial derivative based MPM studies face challenges of underlying instability issues of particle distribution and the complexity of modeling discontinuities. This paper incorporates the state-based peridynamics structure with the MPM to alleviate these problems, which outweighs diferential-based methods in both accuracy and stability. The deviatoric flow theory and a simple yield function are incorporated to animate plasticity

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Integrating Peridynamics to Material Point Method for Modelling Solids and Fracture Dynamics in High Velocity Impact.

The desire for graphical methods to intuitively handle elastoplastic materials has grown hand in hand with the advances made in computer Graphics. Simulating physical materials with dynamic movements to photorealistic resolution is still one of the most crucial and challenging topics, especially involving fractures. Material Point Method (MPM) presents a strong approach for animating elastoplastic materials due to its natural support for arbitrarily large topological deformations and intrinsic collision handling. However, the partial derivative based MPM brings underlying instability issue of handling discontinuous particle distributions and requires computationally expensive treatments to separate broken pieces. The objective of this thesis is to pro- pose a novel MPM solver for robustly and intuitively animating scenarios containing fractures. We are inspired by Peridynamics (PD) which is oriented toward deformations with discontinuities. This study exploits the PD within the MPM scheme to mitigate the difficulties inherent in handling fractures. First, we propose an integral-based MPM by adopting a PD integral energy density function to the MPM weak form and following the standard MPM discretization scheme. Novel elastic, plastic, viscoelastic and fracture models encoding PD bond concepts are designed as constitutive models. The integral-based MPM out-weighs the differential-based MPM in both accuracy and stability. To efficiently model myriad fragments with a MPM solver (especially in high speed impact scenarios), our second contribution is to formulate a rigorous coupling governing equation which integrates the state-based PD within the MPM scheme (Superposition- based MPM) that features an automatic fractures modelling scheme. In SPB-MPM, PD evolves as a result of failure evolution in critical regions while the MPM derives entire problem domain. Giving a low-overhead PD computation to the MPM, this method allows for simulating a breadth of fracture effects, including ductile and brittle fractures. The prominent features at high strain rate in high velocity impact are unattainable through general constitutive models. Our third contribution is to introduce a shock wave effects model and a metallic plastic model which are designed to capture the intricate and characteristic impact behaviours. We simulate a number of representative impact scenarios, including organic fruits, metallic materials and multi-material deformable objects, demonstrating the efficacy of our models