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SUMMARYA novel unstructured remeshing environment for gradient-based shape optimization using triangular finite elements is presented. The remeshing algorithm is based on a truss structure analogy; in solving for the equilibrium position of the truss system, the quadratically convergent Newton's method is used. Exact analytical sensitivity information is therefore made available to the shape optimization algorithm. The overall computational efficiency in gradient-based shape optimization is very high.In solving the truss structure analogy, we compare our quadratically convergent Newton solver with a previously proposed forward Euler solver; this includes notes regarding mesh uniformity, element quality, convergence rates and efficiency.We present three numerical examples; it is then shown that remeshing may introduce discontinuities and local minima. We demonstrate that the effects of these on gradient-based algorithms are alleviated to some extent through mesh refinement, and may largely be overcome with a simple multi-start strategy.

SUMMARYIn topology optimization, it is customary to use reciprocal-like approximations, which result in monotonically decreasing approximate objective functions. In this paper, we demonstrate that efficient quadratic approximations for topology optimization can also be derived, if the approximate Hessian terms are chosen with care. To demonstrate this, we construct a dual SAO algorithm for topology optimization based on a strictly convex, diagonal quadratic approximation to the objective function. Although the approximation is purely quadratic, it does contain essential elements of reciprocal-like approximations: for self-adjoint problems, our approximation is identical to the quadratic or second-order Taylor series approximation to the exponential approximation. We present both a single-point and a two-point variant of the new quadratic approximation.