251 research outputs found
A new class of incremental gradient methods for least squares problems
Caption title.Includes bibliographical references (p. 15-16).Supported by the NSF. 9300494-DMIby Dimitri P. Bertsekas
A Robust Solution Procedure for Hyperelastic Solids with Large Boundary Deformation
Compressible Mooney-Rivlin theory has been used to model hyperelastic solids,
such as rubber and porous polymers, and more recently for the modeling of soft
tissues for biomedical tissues, undergoing large elastic deformations. We
propose a solution procedure for Lagrangian finite element discretization of a
static nonlinear compressible Mooney-Rivlin hyperelastic solid. We consider the
case in which the boundary condition is a large prescribed deformation, so that
mesh tangling becomes an obstacle for straightforward algorithms. Our solution
procedure involves a largely geometric procedure to untangle the mesh: solution
of a sequence of linear systems to obtain initial guesses for interior nodal
positions for which no element is inverted. After the mesh is untangled, we
take Newton iterations to converge to a mechanical equilibrium. The Newton
iterations are safeguarded by a line search similar to one used in
optimization. Our computational results indicate that the algorithm is up to 70
times faster than a straightforward Newton continuation procedure and is also
more robust (i.e., able to tolerate much larger deformations). For a few
extremely large deformations, the deformed mesh could only be computed through
the use of an expensive Newton continuation method while using a tight
convergence tolerance and taking very small steps.Comment: Revision of earlier version of paper. Submitted for publication in
Engineering with Computers on 9 September 2010. Accepted for publication on
20 May 2011. Published online 11 June 2011. The final publication is
available at http://www.springerlink.co
Key issues in computational geomechanics
As stated in the introduction, the three main topics covered in this report are actual research
fields. Different analyses and new developments related with these fields have been presented in
the previous chapters. In the following, after a brief summary of the contributions, some directions
for future research are outlined. Detailed presentations of the conclusions of each contribution are
included in the corresponding sections and subsections.
The most relevant contributions of this report are the following:
1. With respect to the treatment of large boundary displacements:
> Quasistatic and dynamic analyses of the vane test for soft materials using a fluid–based
ALE formulation and different non-newtonian constitutive laws.
> The development of a solid–based ALE formulation for finite strain hyperelastic–plastic
models, with applications to isochoric and non-isochoric cases.
2. Referent to the solution of nonlinear systems of equations in solid mechanics:
> The use of simple and robust numerical differentiation schemes for the computation of
tangent operators, including examples with several non-trivial elastoplastic constitutive
laws.
> The development of consistent tangent operators for substepping time–integration rules,
with the application to an adaptive time–integration scheme.
3. In the field of constitutive modelling of granular materials:
> The efficient numerical modelling of different problems involving elastoplastic models,
including work hardening–softening models for small–strain problems and density–
dependent hyperelastic–plastic models in a large–strain context.
> Robust and accurate simulations of several powder compaction processes, with detailed
analysis of spatial density distributions and verification of the mass conservation principle
Catalyst Acceleration for Gradient-Based Non-Convex Optimization
We introduce a generic scheme to solve nonconvex optimization problems using
gradient-based algorithms originally designed for minimizing convex functions.
Even though these methods may originally require convexity to operate, the
proposed approach allows one to use them on weakly convex objectives, which
covers a large class of non-convex functions typically appearing in machine
learning and signal processing. In general, the scheme is guaranteed to produce
a stationary point with a worst-case efficiency typical of first-order methods,
and when the objective turns out to be convex, it automatically accelerates in
the sense of Nesterov and achieves near-optimal convergence rate in function
values. These properties are achieved without assuming any knowledge about the
convexity of the objective, by automatically adapting to the unknown weak
convexity constant. We conclude the paper by showing promising experimental
results obtained by applying our approach to incremental algorithms such as
SVRG and SAGA for sparse matrix factorization and for learning neural networks
Efficient and accurate approach for powder compaction problems
In this paper, a new approach for powder cold compaction simulations is presented. A density-dependent plastic model within the framework of finite strain multiplicative hyperelastoplasticity is used to describe the highly nonlinear material behaviour; the Coulomb dry friction model is used to capture friction effects at die-powder contact; and an Arbitrary Lagrangian–Eulerian (ALE) formulation is used to avoid the (usual) excessive distortion of Lagrangian meshes caused by large mass fluxes. Several representative examples, involving structured and unstructured meshes are simulated. The results obtained agree with the experimental data and other numerical results reported in the literature. It is shown that, contrary to other Lagrangian and adaptive h-remeshing approaches recently reported for this type of problems, the present approach verifies the mass conservation principle with very low relative errors (less than 1% in all ALE examples and exactly in the pure Lagrangian examples). Moreover, thanks to the use of an ALE formulation and in contrast with other simulations, the presented density distributions do not present spurious oscillations
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