The Hilbert Complex

References

CSFEM for Large Deformation of Solids: From Hilbert

Complexes to Numerical Stability Ali Gerami Matin, Arzhang Angoshtari

Civil and Environmental Engineering

1. Angoshtari, A., Shojaei, M. F., and Yavari, A. (2017).

Compatible-strain mixed finite element methods for 2D

compressible nonlinear elasticity. Computer Methods in

Applied Mechanics and Engineering , 313, 596-631.

2. Arnold, D., Falk, R., and Winther, R. (2010). Finite element

exterior calculus: from Hodge theory to numerical stability.

Bulletin of the American mathematical society , 47(2), 281-

354.

3. Bru, J., and Lesch, M. (1992). Hilbert complexes. Journal

of Functional Analysis , 108(1), 88-132.

4. Gerami Matin, A., Angoshtari, A. (2018). CSFEMs: From

Nonlinear Elasticity Complex to Mixed Finite (In

Progress).

The trial space of the above FEMs for the displacement gradient K satisfy the classical Hadamard jump condition, which is a necessary condition for the compatibility of K. Thus, these mixed FEMs are called compatible-strain mixed FEMs (CSFEMs) for nonlinear elasticity.

Introduction

Implementation

Figure 1. Preliminary results for 2D nonlinear elasticity [1]: The

top row is the standard Cook's membrane problem that

suggests good performance of CSFEMs in bending and in the

near-incompressible regime. The second row is

inhomogeneous compression of a plate; Some other FEMs for

this example suffer from the hourglass instability, but CSFEMs

perform well. The third row shows tension of a plate with a

complex geometry; CSFEMs work well and give accurate

approximations of stresses. The bottom row shows deformed

configurations of a heterogeneous plate under 100% stretch

with different material properties for its inhomogeneity;

CSFEMs provide a convenient framework for modeling

inhomogeneities. Colors in deformed configurations indicate

the distribution of stress, where lighter colors correspond to

larger stresses.

We will numerically implement (1.1), using the High

Performance Computing (HPC). As the first step, we consider

the Laplace equation. This elliptic equation can be used to

study quasi-linear governing equations of large deformations.

FEM Formulation

• The governing equations

• Mixed formulation By using the underlying spaces of the Hilbert complex of nonlinear

elasticity, we introduce the following mixed formulation for nonlinear

elasticity.

• Existing approaches: Mesh-Free Methods (Belytschko et al, 1994), GFEM

(Strouboulis et al, 2000), XFEM (Belytschko et al, 1999), Reduced Integration &

Stabilization (Reese & Wriggers, 2000), Enhanced Strain Methods (Simo & Armero, 1992),

etc.

• Limitations: Bending problems, near-incompressible regime, accurate calculation of

stress, heterogeneous materials, domains with complex geometries, inelastic behaviors,

etc.

The speed up diagram for the parallel implementation of the

Laplace equation shows by using the domain

decomposition approach. The speed up is defined as the

ratio of the uniprocessor execution time over parallel

execution time. Speed up figure implies that the domain

decomposition approach can significantly decreased run-

time near 1/16 (by using 16 processors) in comparison with

the sequential implementing.

• Objective

In comparison with the standard finite element methods for large deformations, CSFEMs will

have more degrees of freedom, i.e., more unknowns per element. Thus, Parallel computation

schemes will be also developed for an efficient implementation of CSFEMs.

plastic deformations

fracture

deformations of

biomaterials

Lar

ge

elas

tic

and

in

elas

tic

def

orm

atio

ns

of

soli

ds

challenge Developing stable numerical methods for

such engineering applications

SEM image of a micro pillar after plastic deformation

(Khalajhedayati, 2015)

Fracture in steel bar

Mechanical Strength of Cornea (Khalajhedayati, 2015)

develop a new class of stable FEMs for large elastic

and inelastic deformations of solids and shells.

Main

tool

Hilbert Complexes

Some preliminary results for 2D

nonlinear elasticity [1] suggest that the proposed numerical

methods will potentially have the following features:

Optimal convergence

rates

free from numerical artifacts

and instabilities

very good performance on domains

with complex geometries

accurate and mesh-

independent approximatio

ns of stress

a convenient framework

for modeling in-

homogeneities

Future works

CSFEM has widespread applications in mechanical

engineering, bioengineering, robotic problems. For

instance; modeling of the soft robots, finding the residual

stress in bio organs that will be significant step for

prediction of the tumor growth.