csfem for large deformation of solids: from hilbert
TRANSCRIPT
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.