Shape Optimization under Uncertainty from a Stochastic by Harald Held PDF

By Harald Held

ISBN-10: 3834809098

ISBN-13: 9783834809094

Optimization difficulties are suitable in lots of parts of technical, commercial, and financial purposes. even as, they pose difficult mathematical examine difficulties in numerical research optimization. Harald Held considers an elastic physique subjected to doubtful inner and exterior forces. in view that easily averaging the prospective loadings will lead to a constitution that will no longer be strong for the person loadings, he makes use of ideas from point set-based form optimization and two-stage stochastic programming. benefiting from the PDE's linearity, he's capable of compute suggestions for an arbitrary variety of situations with out considerably expanding the computational attempt. the writer applies a gradient procedure utilizing the form by-product and the topological gradient to reduce, e.g., the compliance. The stochastic programming viewpoint additionally permits incorporating danger measures into the version that may be extra acceptable aim in lots of sensible functions.

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Extra info for Shape Optimization under Uncertainty from a Stochastic Programming Point of View

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7 for a sketch of a part of a domain which is intersected by the Dirichlet boundary, and the different types of nodes and triangles. (a) Slave nodes in ΘΓD are indicated by squares. Free nodes in Θin are marked by circles. (b) Inner triangles in T in are shown in green. The brown colored triangles belong to T ΓD . Fig. 7: The physical domain is indicated by the blue shaded triangles. The red line is now part of the Dirichlet boundary ΓD . (a) shows a small part of O next to its Dirichlet boundary with marked slave and free nodes.

There are as many basis functions as there are nodes in the grid of triangles, and each of these basis functions takes the value 1 at exactly one node, and 0 at all the other nodes. They are piecewise linear on their support (see Fig. 1). For further details on triangulations, the properties they should have, and other finite elements we refer to [Bra03]. We introduce some notation and basic ingredients for our purposes in the following definition. 1. Let O ⊆ R2 be a polygonal domain. We denote a triangulation of O by T := {τ1 , .

E. F(z) > −∞, if qT + zT W ≥ 0 holds. This can be seen as follows: Suppose, there is a component i ∈ {1, . . , m} with qi + (zT W )i < 0. Then we could define feasible points y(t) := (tδ1i , . . ,tδmi ) ∈ Rm for all t ∈ R,t ≥ 0. Letting t ≥ 0 tend to +∞ would then yield qT + zT W y(t) −→ −∞, and consequently F(z) = −∞. 10), as we want to maximize F(z) in that problem. Hence an optimal z ∈ Rl has to satisfy qT + zT W ≥ 0, or equivalently W T (−z) ≤ q. 25) after replacing1 −z by u. 21); that way there actually are inner and outer minimization problems.

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Shape Optimization under Uncertainty from a Stochastic Programming Point of View by Harald Held

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