Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection

Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations

Restriction methods for shape and topology optimization

This dissertation deals with problems of shape and topology optimization in which the goal is to find the most efficient shape of a physical system. The behavior of this system is captured by the solution to a boundary value problem that in turn depends on the given shape. As such, optimal shape design can be viewed as a form of optimal control in which the control is the shape or domain of the governing state equation. The resulting methodologies have found applications in many areas of engineering, ranging from conceptual layout of high-rise buildings to the design of patient-tailored craniofacial bone replacements. Optimal shape problems and more generally PDE-constrained inverse problems, however, pose several fundamental challenges. For example, these problems are often ill-posed in that they do not admit solutions in the classical sense. The basic compliance minimization problem in structural design, wherein one aims to find the stiffest arrangement of a fixed volume of material, favors non-convergent sequences of shapes that exhibit progressively finer features. To address the ill-posedness, one either enlarges the admissible design space allowing for generalized micro-perforated shapes, an approach known as "relaxation," or alternatively places additional constraints to limit the complexity of the admissible shapes, a strategy commonly referred to as "restriction." We discuss the issue of existence of solutions in detail and outline the key elements of a well-posed restriction formulation for both density and implicit function parametrizations of the shapes. In the latter case, we demonstrate both mathematically and numerically that without an additional "transversality" condition, the usual smearing of the Heaviside map (which links the implicit functions to the governing state equation), no matter how small, will transform the problem into the so-called variable thickness problem, whose theoretical optimal solutions do not have a clearly-defined boundary. Within the restriction setting, we also analyze and provide a justification for the so-called Ersatz approximation in structural optimization where the void regions are filled by a compliant material in order to facilitate the numerical implementation. Another critically important but challenging aspect of optimal shape design is dealing with the resulting large-scale non-convex optimization systems which contain many local minima and require expensive function evaluations and gradient calculations. As such, conventional nonlinear programming methods may not be adequately efficient or robust. We develop a simple and tailored optimization algorithm for solving structural topology optimization problems with an additive regularization term and subject only to a set of box constraints. The proposed splitting algorithm matches the structure of the problem and allows for separate treatment of the cost function, the regularizer, and the constraints. Though our mathematical and numerical investigation is mainly focused on Tikhonov regularization, one important feature of the splitting framework is that it can accommodate nonsmooth regularization schemes such as total variation penalization. We also investigate the use of isoparametric polygonal finite elements for the discretization of the design and response fields in two-dimensional topology optimization problems. We show that these elements, unlike their low-order Lagrangian counterparts, are not susceptible to certain grid-scale instabilities (e.g., checkerboard patterns) that may appear as a result of inaccurate analysis of the design response. The better performance of polygonal discretizations is attributed to the enhanced approximation characteristics of these elements, which also alleviate shear and volumetric locking phenomena. In regards to the latter property, we demonstrate that low-order finite element spaces obtained from polygonal discretizations satisfy the well-known Babuska-Brezzi condition required for stability of the mixed variational formulation of incompressible elasticity and Stokes flow problems. Conceptually, polygonal finite elements are the natural extension of commonly used linear triangles and bilinear quads to all convex n-gons. To facilitate their use, we present a simple but robust meshing algorithm that utilizes Voronoi diagrams to generate convex polygonal discretizations of implicit geometries. Finally, we provide a self-contained discretization and analysis Matlab code using polygonal elements, along with a general framework for topology optimization

New Advances in Topology Optimization

IASS-IACM 2008 Session: New Advances in Topology Optimization -- Session Organizers: Glaucio H. PAULINO (UIUC), Emilio SILVA (University of Sao Paulo) -- Keynote Lecture: "Topology optimization with adaptive mesh refinement" by Eric DE STURLER (Virginia Tech), Glaucio H. PAULINO, Shun WANG (University of Illinois at Urbana-Champaign) -- "Strategies for computational efficiency in continuum structural topology optimization of sparse 3D systems" by Colby C. SWAN , Salam F. RAHMATALLA (University of Iowa) -- "Wachpress elements for topology optimization" by Cameron TALISCHI , Glaucio H. PAULINO, Chau H. LE (University of Illinois at Urbana-Champaign) -- "Topology optimization technique considering both static and dynamic characteristics of the structures" by S. J. LEE , J. E. BAE (Gyeongsang National University) -- Keynote Lecture: "Topology optimization method utilizing iterative solvers with subspace recycling applied to high-resolution electrical impedance tomography" by Luis Augusto Motta MELLO , Emilio Carlos Nelli SILVA (University of Sao Paulo), Eric DE STURLER (Virginia Tech), Glaucio H. PAULINO (University of Illinois at Urbana-Champaign) -- "Topology optimization considering fabrication errors and length scale constraints" by James K. GUEST (Johns Hopkins University) -- "A simple and effective inverse projection scheme for void distribution control in topology optimization" by Glaucio H. PAULINO (University of Illinois at Urbana-Champaign), Sylvia ALMEIDA (Universidade Federal de Goias), Emilio Carlos Nelli SILVA (University of Sao Paulo) -- "Design of dynamic laminate piezoelectric sensors and actuators using topology optimization" by Paulo Henrique NAKASONE , Emilio Carlos Nelli SILVA (University of Sao Paulo