This paper introduces a method for variabletopology shape optimization of elastic structures called the perimeter method. An upper-bound constraint on the perimeter of the solid part of the structure ensures a well-posed design problem. The perimeter constraint allows the designer to control the number of holes in the optimal design and to establish their characteristic length scale. Finite element implementations generate practical designs that are convergent with respect to grid refinement. Thus, an arbitrary level of geometric resolution can be achieved, so single-step procedures for topology design and detailed shape design are possible. The perimeter method eliminates the need for relaxation, thereby circumventing many of the complexities and restrictions of other approaches to topology design.and Strang 1986). In particular, we typically can construct a nonconvergent sequence such that the compliance reduces monotonically. Therefore, an optimal design does not exist.

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This paper deals with the simultaneous optimization of material and structure for minimum compliance. Material properties are represented in the most general form possible for a (locally) linear elastic continuum, namely the unrestricted set of elements of positive semi-definite constitutive tensors and cost measures based on certain invariants of the tensors. Analytical forms are derived for the optimized material properties. These results, which apply in general, indicate that the optimized material is orthotropic with the directions of orthotropy following the directions of principal strains. The analysis for optimization of the material leads to a reduced structural optimization problem, for which the existence of solutions can be shown and for which effective methods for computational solution can be devised.

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SUMMARYWe present a new method for shape optimization that uses an analytical description of the varying design geometry as the control in the optimization problem. A straightforward filtering technique projects the design geometry onto a fictitious analysis domain to support simplified response and sensitivity analysis. However, the analytical geometry model is referenced directly for all purely geometric calculations. The method thus combines the advantages of direct geometry representations with the simplified analysis procedures that are possible with fictitious domain analysis methods, such as the material distribution methods commonly used in topology optimization. The projected geometry measure converges to the indicator function of the analytical geometry model in the limit of numerical mesh refinement. Consequently, optimal designs obtained with the new method converge to solutions of well-defined continuum optimization problems in the limit of mesh refinement. This property is confirmed in example computations for minimum compliance design of an elastic structure subject to a volume constraint and for minimum volume design subject to a maximum stress constraint.

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We propose a new algorithm for constructing finite-element meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain Ω and a target time value T , our method constructs a tetrahedral mesh of the spacetime domain Ω × [0, T ] in constant running time per tetrahedron in IR 3 using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.