56 research outputs found

    Nonlinear hyperelasticity-based mesh optimisation

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    In this work, various aspects of PDE-based mesh optimisation are treated. Different existing methods are presented, with the focus on a class of nonlinear mesh quality functionals that can guarantee the orientation preserving property. This class is extended from simplex to hypercube meshes in 2d and 3d. The robustness of the resulting mesh optimisation method allows the incorporation of unilateral boundary conditions of place and r-adaptivity with direct control over the resulting cell sizes. Also, alignment to (implicit) surfaces is possible, but in all cases, the resulting functional is hard to treat analytically and numerically. Using theoretical results from mathematical elasticity for hyperelastic materials, the existence and non-uniqueness of minimisers can be established. This carries over to the discrete case, for the solution of which tools from nonlinear optimisation are used. Because of the considerable numerical effort, a class of linear preconditioners is developed that helps to speed up the solution process

    Computation and optimal perturbation of finite-time coherent sets for aperiodic flows without trajectory integration

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    Understanding the macroscopic behavior of dynamical systems is an important tool to unravel transport mechanisms in complex flows. A decomposition of the state space into coherent sets is a popular way to reveal this essential macroscopic evolution. To compute coherent sets from an aperiodic time-dependent dynamical system we consider the relevant transfer operators and their infinitesimal generators on an augmented space-time manifold. This space-time generator approach avoids trajectory integration, and creates a convenient linearization of the aperiodic evolution. This linearization can be further exploited to create a simple and effective spectral optimization methodology for diminishing or enhancing coherence. We obtain explicit solutions for these optimization problems using Lagrange multipliers and illustrate this technique by increasing and decreasing mixing of spatial regions through small velocity field perturbations

    Quarklets: Construction and Application in Adaptive Frame Methods

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    This thesis is concerned with the construction and application of a new class of functions called quarklets. With the intention of constructing an adaptive hp-method based on wavelets, they do arise out of the latter through an enrichment with polynomials. The starting point for the construction is the real axis. There, we derive frames for the Sobolev space H^s(R). Through a boundary adaptation, tensorization and the application of a scale-dependent extension operator we are even able to construct quarklet frames on very general domains in multiple spatial dimensions. With these frames at hand we can discretize linear elliptic operator equations in a stable way. The discrete system can be handled with an adaptive numerical scheme. For this purpose it is necessary to show the compressibility of the stiffness matrix. We do this for the prototypical example of the Poisson equation independently of the dimension and in this way we are able to prove the optimality of the adaptive scheme. By the latter we mean that the approximation rate of the best n-term quarklet approximation is realized by the scheme. Finally, we carry out some numerical experiments in one and two spatial dimensions, where the theoretical findings are validated in practice and moreover, the value of the quarklets in the numerical scheme becomes visible by analysing the quarklet coefficients of the approximate solutions

    The Structure of Isoperimetric Bubbles on Rn\mathbb{R}^n and Sn\mathbb{S}^n

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    The multi-bubble isoperimetric conjecture in nn-dimensional Euclidean and spherical spaces from the 1990's asserts that standard bubbles uniquely minimize total perimeter among all q1q-1 bubbles enclosing prescribed volume, for any qn+2q \leq n+2. The double-bubble conjecture on R3\mathbb{R}^3 was confirmed in 2000 by Hutchings-Morgan-Ritor\'e-Ros, and is nowadays fully resolved for all n2n \geq 2. The double-bubble conjecture on S2\mathbb{S}^2 and triple-bubble conjecture on R2\mathbb{R}^2 have also been resolved, but all other cases are in general open. We confirm the conjecture on Rn\mathbb{R}^n and on Sn\mathbb{S}^n for all qmin(5,n+1)q \leq \min(5,n+1), namely: the double-bubble conjectures for n2n \geq 2, the triple-bubble conjectures for n3n \geq 3 and the quadruple-bubble conjectures for n4n \geq 4. In fact, we show that for all qn+1q \leq n+1, a minimizing cluster necessarily has spherical interfaces, and after stereographic projection to Sn\mathbb{S}^n, its cells are obtained as the Voronoi cells of qq affine-functions, or equivalently, as the intersection with Sn\mathbb{S}^n of convex polyhedra in Rn+1\mathbb{R}^{n+1}. Moreover, the cells (including the unbounded one) are necessarily connected and intersect a common hyperplane of symmetry, resolving a conjecture of Heppes. We also show for all qn+1q \leq n+1 that a minimizer with non-empty interfaces between all pairs of cells is necessarily a standard bubble. The proof makes crucial use of considering Rn\mathbb{R}^n and Sn\mathbb{S}^n in tandem and of M\"obius geometry and conformal Killing fields; it does not rely on establishing a PDI for the isoperimetric profile as in the Gaussian setting, which seems out of reach in the present one.Comment: 90 pages, 14 figures. Incorporated comments by referee, added Figure 5 regarding quasi-centers, and Appendix B with a technical computation. Final version, to appear in Acta Mat

    [Activity of Institute for Computer Applications in Science and Engineering]

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    This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science
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