The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation

We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one and two-dimension problems confirm the convergence rates of the theoretical results.Comment: 22 pages, 4 figure

3D Virtual Elements for Elastodynamic Problems

A virtual element framework for nonlinear elastodynamics is outlined within this work. The virtual element method (VEM) can be considered as an extension of the classical finite element method. While the finite element method (FEM) is restricted to the usage of regular shaped elements, VEM allows to use non-convex shaped elements for the spatial discretization [1]. It has been applied to various engineering problems in elasticity and other areas, such as plasticity or fracture mechanics as outlined in [3, 4]. This work deals with the extension of VEM to dynamic problems. Low-order ansatz functions in two and three dimensions, with elements being arbitrary shaped, are used in this contribution. The formulations considered in this framework are based on minimization of energy, where a pseudo potential is used for the dynamic behavior. While the stiffness-matrix needs a suitable stabilization, the mass-matrix can be calculated fully through the projection part. For the implicit time integration, Newmark-Method is used. To show the performance of the method, various numerical examples in 2D and 3D are presented

The Application of Sequential Convex Programming to Large-Scale Structural Optimization Problems

Structural design problems are often modeled using finite element methods. Such models are often characterized by constraint functions that are not explicitly defined in terms of the design variables. These functions are typically evaluated through numerical finite element analysis (FEA). Optimizing large-scale structural design models requires computationally expensive FEAs to obtain function and gradient values. An optimization approach which uses the SCP sequential convex programming algorithm of Zillober, integrated as the optimizer in the Automated Structural Optimization System (ASTROS), is tested. The traditional approach forms an explicitly defined approximate subproblem at each design iteration that is solved using the method of modified feasible directions. In an alternative approach, the SCP subroutine is called to formulate and solve the approximate subproblem. The SCP method is an implementation of the Method of Moving Asymptotes algorithm with five different asymptote determination strategies. This study investigates the effect of different asymptote determination strategies and constraint retention strategies on computational efficiency. The approach is tested on three large-scale structural design models, including one with constraints from multiple disciplines. Results and comparisons to the traditional approach are given. The largest of the three models, which had 1527 design variables and 6124 constraints, was solved to optimality with ASTROS for the first time using a mathematical programming method. The structural weight of the resulting design is 9% lower than the previously recorded minimum weight

State update algorithm for associative elastic-plastic pressure-insensitive materials by incremental energy minimization

This work presents a new state update algorithm for small-strain associative elastic-plastic constitutive models, treating in a unified manner a wide class of deviatoric yield functions with linear or nonlinear strain-hardening. The algorithm is based on an incremental energy minimization approach, in the framework of generalized standard materials with convex free energy and dissipation potential. An efficient method for the computation of the latter, its gradient and its Hessian is provided, using Haigh-Westergaard stress invariants. Numerical results on a single material point loading history and finite element simulations are reported to prove the effectiveness and the versatility of the method. Its merit turns out to be complementary to the classical return map strategy, because no convergence difficulties arise if the stress is close to high curvature points of the yield surface