Fast Corotated Elastic SPH Solids with Implicit Zero-Energy Mode Control

2021 ◽  
Vol 4 (3) ◽  
pp. 1-21
Author(s):  
Tassilo Kugelstadt ◽  
Jan Bender ◽  
José Antonio Fernández-Fernández ◽  
Stefan Rhys Jeske ◽  
Fabian Löschner ◽  
...  
Keyword(s):  
Operator Splitting ◽  
Computation Time ◽  
Elastic Solid ◽  
New Physics ◽  
Integration Step ◽  
Splitting Scheme ◽  
System Matrix ◽  
Zero Energy ◽  
Elastic Solids ◽  
Particle Distributions

We develop a new operator splitting formulation for the simulation of corotated linearly elastic solids with Smoothed Particle Hydrodynamics (SPH). Based on the technique of Kugelstadt et al. [2018] originally developed for the Finite Element Method (FEM), we split the elastic energy into two separate terms corresponding to stretching and volume conservation, and based on this principle, we design a splitting scheme compatible with SPH. The operator splitting scheme enables us to treat the two terms separately, and because the stretching forces lead to a stiffness matrix that is constant in time, we are able to prefactor the system matrix for the implicit integration step. Solid-solid contact and fluid-solid interaction is achieved through a unified pressure solve. We demonstrate more than an order of magnitude improvement in computation time compared to a state-of-the-art SPH simulator for elastic solids. We further improve the stability and reliability of the simulation through several additional contributions. We introduce a new implicit penalty mechanism that suppresses zero-energy modes inherent in the SPH formulation for elastic solids, and present a new, physics-inspired sampling algorithm for generating high-quality particle distributions for the rest shape of an elastic solid. We finally also devise an efficient method for interpolating vertex positions of a high-resolution surface mesh based on the SPH particle positions for use in high-fidelity visualization.

scholarly journals Operator splitting around Euler-Maruyama scheme and high order discretization of heat kernels

2020 ◽  
Author(s):  
Toshihiro Yamada ◽  
Yuga Iguchi
Keyword(s):  
Commutative Algebra ◽  
Computational Algorithm ◽  
Diffusion Processes ◽  
Operator Splitting ◽  
Heat Kernels ◽  
Heat Equations ◽  
Discretization Method ◽  
Splitting Scheme ◽  
Order Operator ◽  
Numerical Examples

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker-Campbell-Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler-Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

Finite strains in an anisotropic elastic continuum

1950 ◽  
Vol 202 (1070) ◽  
pp. 345-358 ◽  
Keyword(s):  
Equations Of State ◽  
Elastic Solid ◽  
Finite Strains ◽  
Free Energy Function ◽  
Elastic Solids ◽  
Elastic Continuum ◽  
Invariance Properties ◽  
Anisotropic Elastic ◽  
Necessary And Sufficient ◽  
Thermodynamic Restrictions

The discussion in a previous paper (Oldroyd 1950), on the invariance properties required of the equations of state of a homogeneous continuum, is extended by taking into account thermodynamic restrictions on the form of the equations, in the case of an elastic solid deformed from an unstressed equilibrium configuration. The general form of the finite strainstress-temperature relations, expressed in terms of a free-energy function, is deduced without assuming that the material is isotropic. The results of other authors based on the assumption of isotropy are shown to follow as particular cases. The equations of state are derived by considering quasi-static changes in an elastic solid continuum; the results then apply to non-ideally elastic solids in equilibrium, or subjected to quasi-static changes only, and to ideally elastic solids in general motion. A necessary and sufficient compatibility condition for the finite strains at different points of a continuum is also derived. As a simple illustration of the derivation and use of equations of state involving anisotropic physical constants, the torsion of an anisotropic cylinder is discussed briefly.

scholarly journals Embedded Zassenhaus Expansion to Splitting Schemes: Theory and Multiphysics Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jürgen Geiser
Keyword(s):  
Fluid Dynamics ◽  
Convergence Rates ◽  
Operator Splitting ◽  
Main Idea ◽  
Computation Time ◽  
Product Formula ◽  
Splitting Methods ◽  
Splitting Schemes ◽  
Operator Splitting Methods ◽  
Multiphysics Problems

We present some operator splitting methods improved by the use of the Zassenhaus product and designed for applications to multiphysics problems. We treat iterative splitting methods that can be improved by means of the Zassenhaus product formula, which is a sequential splitting scheme. The main idea for reducing the computation time needed by the iterative scheme is to embed fast and cheap Zassenhaus product schemes, since the computation of the commutators involved is very cheap, since we are dealing with nilpotent matrices. We discuss the coupling ideas of iterative and sequential splitting techniques and their convergence. While the iterative splitting schemes converge slowly in their first iterative steps, we improve the initial convergence rates by embedding the Zassenhaus product formula. The applications are to multiphysics problems in fluid dynamics. We consider phase models in computational fluid dynamics and analyse how to obtain higher order operator splitting methods based on the Zassenhaus product. The computational benefits derive from the use of sparse matrices, which arise from the spatial discretisation of the underlying partial differential equations. Since the Zassenhaus formula requires nearly constant CPU time due to its sparse commutators, we have accelerated the iterative splitting schemes.

The Linear Magnetoelastic Problem for a Soft Ferromagnetic Elastic Solid With a Finite Crack

1977 ◽  
Vol 44 (1) ◽  
pp. 47-50 ◽  
Author(s):  
Y. Shindo
Keyword(s):  
Integral Transform ◽  
Elastic Solid ◽  
Elastic Solids ◽  
Infinite Body ◽  
Magnetostatic Field ◽  
Multidomain Structure ◽  
Soft Ferromagnetic ◽  
Finite Crack ◽  
Integral Transform Technique ◽  
Maxwell Stresses

Following a linear theory for the soft ferromagnetic elastic materials of multidomain structure, the distribution of magnetoelastic stresses and Maxwell stresses in an infinite body with a finite crack permeated by an uniform magnetostatic field normal to the crack surfaces is investigated. The soft ferromagnetic elastic solids with a finite crack are considered to be composed of materials with isotropic, cubic, or uniaxial symmetry. A solution for the infinite solid is obtained by the use of integral transform technique. The magnetoelastic stresses and the Maxwell stresses are expressed in closed forms.

scholarly journals Existence of solutions of plane traction problems for inextensible transversely isotropic elastic solids

1975 ◽  
Vol 19 (1) ◽  
pp. 40-54 ◽  
Author(s):  
L. W. Morland
Keyword(s):  
Existence Of Solutions ◽  
Lateral Displacement ◽  
Elastic Solid ◽  
Transversely Isotropic ◽  
Elastic Solids ◽  
Standard Potential ◽  
Linear Elastic ◽  
Axis Of Symmetry ◽  
Uniqueness And Existence ◽  
Traction Boundary Conditions

AbstractA plane strain or plane stress configuration of an inextensible transversely isotropic linear elastic solid with the axis of symmetry in the plane, leads to a harmonic lateral displacement field in stretched coordinates. Various displacement and mixed displacement-traction boundary conditions yield standard boundary-value problems of potential theory for which uniqueness and existence of solutions are well established. However, the important case of prescribed tractions at each boundary point gives a non-standard potential problem involving linking of boundary values at opposite ends of chords parallel to the axis of material symmetry. Uniqueness and existence of solutions, within arbitrary rigid motions, are now established for the traction problem for general domains.

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