Ivan F. M. Menezes

Nonlinear Finite Element Analysis using an Object-Oriented Philosophy – Application to Beam Elements and to the Cosserat Continuum

Abstract.An Object-Oriented Programming (OOP) frame-work is presented for solving nonlinear structural mechanics problems by means of the Finite Element Method (FEM). Emphasis is placed on engineering applications (geometrically nonlinear beam model, and elastoplastic Cosserat continuum), and OOP is employed as an effective tool, which plays an important role in the FEM treatment of such applications. The implementation is based on computational abstractions of both mathematical and physical concepts associated to structural mechanics problems involving geometrical and material nonlinearities. The overall class organization for nonlinear mechanics modeling is discussed in detail. All the analyses rely on a generic control class where several classical and modern nonlinear solution schemes are available. Examples which explore, demonstrate and validate the main features of the overall computational system are presented and discussed.

Topology optimization using polytopes

Abstract Meshing complex engineering domains is a challenging task. Arbitrary polyhedral meshes can provide the much needed flexibility in automated discretization of such domains. The geometric property of polyhedral meshes such as its unstructured nature and the connectivity of faces between elements makes them specially attractive for topology optimization applications. Numerical anomalies in designs such as the single node connections and checkerboard pattern can be naturally circumvented with polyhedrons. In the current work, we solve the governing three-dimensional elasticity state equation using the Virtual Element Method (VEM) approach. The main characteristic difference between VEM and standard finite element methods (FEM) is that in VEM the canonical basis functions are not constructed explicitly. Rather the stiffness matrix is computed directly utilizing a projection map which extracts the linear component of the deformation. Such a construction guarantees the satisfaction of the patch test (used by engineers as an indicator of optimal convergence of numerical solutions under mesh refinement). Finally, the computations reduce to the evaluation of matrices which contain purely geometric surface facet quantities. The present work focuses on the first-order VEM in which the degrees of freedom are associated with the vertices. The features of the current optimization approach are demonstrated using numerical examples for compliance minimization and compliant mechanism problems.

A Unified Library of Nonlinear Solution Schemes

1 A Unified Library of Nonlinear Solution Schemes Sofie E. Leon1, Glaucio H. Paulino1, Anderson Pereira2, Ivan F. M. Menezes2, Eduardo N. Lages2 1 Civil and Environmental Engineering Department, University of Illinois, Urbana, IL, USA 2 Group of Technology in Computer Graphics, Pontifical Catholic University ,Rio de Janeiro, RJ, Brazil 3 Center of Technology, Federal University of Alagoas, Maceio, Alagoas, Brazil

An object-oriented framework for finite element analysis based on a compact topological data structure

This paper describes an ongoing work in the development of a finite element analysis system, called TopFEM, based on the compact topological data structure, TopS [1,2]. This new framework was written to take advantage of the topological data structure together with object-oriented programming concepts to handle a variety of finite element problems, spanning from fracture mechanics to topology optimization, in an efficient, but generic fashion. The class organization of the TopFEM system is described and discussed within the context of other frameworks in the literature that share similar ideas, such as GetFEM++, deal.II, FEMOOP and OpenSees. Numerical examples are given to illustrate the capabilities of TopS attached to a finite element framework in the context of fracture mechanics and to establish a benchmark with other implementations that do not make use of a topological data structure.

Otimização Topológica considerando Análise Limite

Resumo. Este trabalho apresenta uma formulação puramente baseada em plasticidade para ser aplicada à otimização topológica. A principal ideia da otimização topológica em mecânica dos sólidos é encontrar a distribuição de material dentro do domı́nio de forma a otimizar uma medida de performance e satisfazer um conjunto de restrições. Uma possibilidade é minimizar a flexibilidade da estrutura satisfazendo que o volume seja menor do que um determinado valor. Essa é a formulação clássica da otimização topológica, que é vastamente utilizada na literatura. Não obstante fornecer resultados interessantes, verificações adicionais, relativas ao critério de segurança estrutural, são necessárias para viabilizar sua aplicação prática. Esse critério de segurança pode ser definido como limitar as tensões elásticas ao critério de plastificação em todo o domı́nio. Esta definição leva à otimização topológica com restrições de tensões. Por outro lado, em aplicações de projeto por estado limite último busca-se tirar proveito de toda capacidade resistente da estrutura. Dessa forma, este trabalho aborda a incorporação do projeto estrutural plástico à otimização topológica. A formulação proposta é uma extensão da análise limite, que fornece uma estimativa da carga de colapso de uma estrutura diretamente por meio da programação matemática, assegurando a eficiência computacional da metodologia proposta. De forma a verificar a otimização topológica plástica e comparar a topologia final com as obtidas através da otimização topológica clássica e da com restrição de tensões, são apresentados exemplos numéricos.

Robust Topology Optimization under Uncertain Loads - A Spectral Stochastic Approach

Abstract. A spectral stochastic approach for structural topology optimization in the presence of uncertainties in the magnitude and direction of the applied loads is proposed. The application of this approach in the representation and propagation of uncertainties presents a low computational cost compared to classical techniques, such as Monte Carlo simulation. A recent development of spectral representation methods, known as generalized polynomial chaos (gPC), has become one the most widely used methods by exhibiting fast convergence when the solution depends smoothly on the random parameters. Therefore, in this work, gPC is applied to estimate the statistical measures of the compliance of 2D continuum structures, which we call Robust Topology Optimization. To demonstrate the accuracy and applicability of the proposed method, we solve robust topology optimization where we minimize the influence of stochastic variability on the mean design. Representative examples of topology optimization of continuum structures under load uncertainties are presented. The results demonstrate that load uncertainties play an important role in the optimal design. It is also shown that results obtained from the gPC method are in excellent agreement with those obtained from Monte Carlo simulation.