222 research outputs found
Geometric partial differential equations: Surface and bulk processes
The workshop brought together experts representing a wide range of topics in geometric partial differential equations ranging from analyis over numerical simulation to real-life applications. The main themes of the conference were the analysis of curvature energies, new developments in pdes on surfaces and the treatment of coupled bulk/surface problems
Isogeometric Approximation of Variational Problems for Shells
The interaction of applied geometry and numerical simulation is a growing field in the interplay of com- puter graphics, computational mechanics and applied mathematics known as isogeometric analysis. In this thesis we apply and analyze Loop subdivision surfaces as isogeometric tool because they provide great flexibility in handling surfaces of arbitrary topology combined with higher order smoothness. Compared with finite element methods, isogeometric methods are known to require far less degrees of freedom for the modeling of complex surfaces but at the same time the assembly of the isogeo- metric matrices is much more time-consuming. Therefore, we implement the isogeometric subdivision method and analyze the experimental convergence behavior for different quadrature schemes. The mid-edge quadrature combines robustness and efficiency, where efficiency is additionally increased via lookup tables. For the first time, the lookup tables allow the simulation with control meshes of arbitrary closed connectivity without an initial subdivision step, i.e. triangles can have more than one vertex with valence different from six. Geometric evolution problems have many applications in material sciences, surface processing and modeling, bio-mechanics, elasticity and physical simulations. These evolution problems are often based on the gradient flow of a geometric energy depending on first and second fundamental forms of the surface. The isogeometric approach allows a conforming higher order spatial discretization of these geometric evolutions. To overcome a time-error dominated scheme, we combine higher order space and time discretizations, where the time discretization based on implicit Runge-Kutta methods. We prove that the energy diminishes in every time-step in the fully discrete setting under mild time-step restrictions which is the crucial characteristic of a gradient flow. The overall setup allows for a general type of fourth-order energies. Among others, we perform experiments for Willmore flow with respect to different metrics. In the last chapter of this thesis we apply the time-discrete geodesic calculus in shape space to the space of subdivision shells. By approximating the squared Riemannian distance by a suitable energy, this approach defines a discrete path energy for a consistent computation of geodesics, logarithm and exponential maps and parallel transport. As approximation we pick up an elastic shell energy, which measures the deformation of a shell by membrane and bending contributions of its mid-surface. BĂ©zier curves are a fundamental tool in computer-aided geometric design. We extend these to the subdivision shell space by generalizing the de Casteljau algorithm. The evaluation of BĂ©zier curves depends on all input data. To solve this problem, we introduce B-splines and cardinal splines in shape space by gluing together piecewise BĂ©zier curves in a smooth way. We show examples of quadratic and cubic BĂ©zier curves, quadratic and cubic B-splines as well as cardinal splines in subdivision shell space
Geometric partial differential equations: Theory, numerics and applications
This workshop concentrated on partial differential equations involving stationary and evolving surfaces in which geometric quantities play a major role. Mutual interest in this emerging field stimulated the interaction between analysis, numerical solution, and applications
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Mini-Workshop: Mathematics of Biological Membranes
[no abstract available
A convergent evolving finite element algorithm for Willmore flow of closed surfaces
A proof of convergence is given for a novel evolving surface finite element
semi-discretization of Willmore flow of closed two-dimensional surfaces, and
also of surface diffusion flow. The numerical method proposed and studied here
discretizes fourth-order evolution equations for the normal vector and mean
curvature, reformulated as a system of second-order equations, and uses these
evolving geometric quantities in the velocity law interpolated to the finite
element space. This numerical method admits a convergence analysis in the case
of continuous finite elements of polynomial degree at least two. The error
analysis combines stability estimates and consistency estimates to yield
optimal-order -norm error bounds for the computed surface position,
velocity, normal vector and mean curvature. The stability analysis is based on
the matrix--vector formulation of the finite element method and does not use
geometric arguments. The geometry enters only into the consistency estimates.
Numerical experiments illustrate and complement the theoretical results
Second-order Shape Optimization for Geometric Inverse Problems in Vision
We develop a method for optimization in shape spaces, i.e., sets of surfaces
modulo re-parametrization. Unlike previously proposed gradient flows, we
achieve superlinear convergence rates through a subtle approximation of the
shape Hessian, which is generally hard to compute and suffers from a series of
degeneracies. Our analysis highlights the role of mean curvature motion in
comparison with first-order schemes: instead of surface area, our approach
penalizes deformation, either by its Dirichlet energy or total variation.
Latter regularizer sparks the development of an alternating direction method of
multipliers on triangular meshes. Therein, a conjugate-gradients solver enables
us to bypass formation of the Gaussian normal equations appearing in the course
of the overall optimization. We combine all of the aforementioned ideas in a
versatile geometric variation-regularized Levenberg-Marquardt-type method
applicable to a variety of shape functionals, depending on intrinsic properties
of the surface such as normal field and curvature as well as its embedding into
space. Promising experimental results are reported
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