Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes

In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the latter remains bounded and the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed multilevel solvers are shown to be convergent in practice, even when some of the theoretical assumptions are not fully satisfied

Contributions to handle maximum size constraints in density-based topology optimization

peer reviewedThe maximum size formulation in topology optimization restricts the amount of material within a test region in each point in the design domain, leading to a highly constrained problem. In this work the local constraints are aggregated into a single one by p-mean and p-norm functions, classically used for stress constraints. Moreover, a new test region is investigated which is a ring instead of the classical circle around the element. These developments were implemented for compliance minimization with the MBB beam test case. Results indicate that p-mean performs better in the maximum size field than p-norm, because it underestimates the most violated constraint. This gives some relaxation to the problem that allows stiffer connections. Similar effect has been observed for the ring-shaped region which reduces the amount of holes that are introduced in the structure, specially in the connection of solid members. In addition, it is shown that the maximum size formulation allows the definition of the minimum gap between solid members which gives designers more control over the geometry. The developments have been illustrated and validated with compliance minimization tests of 2D-domains.Aero

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