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