@misc{SeebassBeckGellermannetal., author = {Seebass, Martin and Beck, Rudolf and Gellermann, Johanna and Nadobny, Jacek and Wust, Peter}, title = {Electromagnetic phased arrays for regional hyperthermia -- optimal frequency and antenna arrangement}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5961}, number = {00-28}, abstract = {In this paper we investigate the effects of the three-dimensional arrangement of antennas and frequency on temperature distributions that can be achieved in regional hyperthermia using an electromagnetic phased array. We compare the results of power-based and temperature-based optimization. Thus we are able to explain the discrepancies between previous studies favouring more antenna rings on the one hand and more antennas per ring on the other hand. We analyze the sensitivity of the results with respect to changes in amplitudes and phases as well as patient position. This analysis can be used for different purposes. First, it provides additional criteria for selecting the optimal frequency. Second, it can be used for specifying the required phase and amplitude accuracy for a real phased array system. Furthermore, it may serve as a basis for technological developments in order to reduce both types of sensitivities described above.}, language = {en} } @misc{BurgerKloseSchaedleetal., author = {Burger, Sven and Klose, Roland and Sch{\"a}dle, Achim and Zschiedrich, Lin}, title = {HelmPole - A finite element solver for scattering problems on unbounded domains: Implementation based on PML}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7609}, number = {03-38}, abstract = {The solution of scattering problems described by the Helmholtz equation on unbounded domains is of importance for a wide variety of applications, for example in electromagnetics and acoustics. An implementation of a solver for scattering problems based on the programming language Matlab is introduced. The solver relies on the finite-element-method and on the perfectly-matched-layer-method, which allows for the simulation of scattering problems on complex geometries surrounded by inhomogeneous exterior domains. This report gives a number of detailed examples and can be understood as a user manual to the freely accessible code of the solver HelmPole.}, language = {en} } @phdthesis{Schmidt, author = {Schmidt, Frank}, title = {A New Approach to Coupled Interior-Exterior Helmholtz-Type Problems: Theory and Algorithms}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7128}, number = {02-45}, abstract = {The work presents a new approach to the numerical solution of time-harmonic and time-dependent scattering problems. We replace Sommerfeld's radiation condition valid for the Helmholtz equation by a more general concept called pole condition. The pole condition is based on the Laplace transform of the exterior solution and allows a characterization of outgoing waves. Both new insight into the analysis of scattering problems as well as new numerical algorithms are obtained.}, language = {en} } @misc{WeiserSchielaDeuflhard, author = {Weiser, Martin and Schiela, Anton and Deuflhard, Peter}, title = {Asymptotic Mesh Independence of Newton's Method Revisited}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7352}, number = {03-13}, abstract = {The paper presents a new affine invariant theory on asymptotic mesh independence of Newton's method in nonlinear PDEs. Compared to earlier attempts, the new approach is both much simpler and more natural from the algorithmic point of view. The theory is exemplified at collocation methods for ODE boundary value problems and at finite element methods for elliptic PDE problems.}, language = {en} } @misc{WeiserDeuflhardErdmann, author = {Weiser, Martin and Deuflhard, Peter and Erdmann, Bodo}, title = {Affine conjugate adaptive Newton methods for nonlinear elastomechanics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7768}, number = {04-01}, abstract = {The paper extends affine conjugate Newton methods from convex to nonconvex minimization, with particular emphasis on PDE problems originating from compressible hyperelasticity. Based on well-known schemes from finite dimensional nonlinear optimization, three different algorithmic variants are worked out in a function space setting, which permits an adaptive multilevel finite element implementation. These algorithms are tested on two well-known 3D test problems and a real-life example from surgical operation planning.}, language = {en} } @misc{ZschiedrichKloseSchaedleetal., author = {Zschiedrich, Lin and Klose, Roland and Sch{\"a}dle, Achim and Schmidt, Frank}, title = {A new Finite Element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D}, doi = {10.1016/j.cam.2005.03.047}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7662}, number = {03-44}, abstract = {In this paper we propose a new finite element realization of the Perfectly Matched Layer method (PML-method). Our approach allows to deal with arbitrary shaped polygonal domains and with certain types of inhomogeneous exterior domains. Among the covered inhomogeneities are open waveguide structures playing an essential role in integrated optics. We give a detailed insight to implementation aspects. Numerical examples show exponential convergence behavior to the exact solution with the thickness of the PML sponge layer.}, language = {en} } @misc{OevermannScharfenbergKlein, author = {Oevermann, Michael and Scharfenberg, Carsten and Klein, Rupert}, title = {A sharp interface finite volume method for elliptic equations on Cartesian grids}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10900}, number = {08-38}, abstract = {We present a second order sharp interface finite volume method for the solution of the three-dimensional poisson equation with variable coefficients on Cartesian grids. In particular, we focus on interface problems with discontinuities in the coefficient, the source term, the solution, and the fluxes across the interface. The method uses standard piecewiese trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27-point stencil. Singularities associated with vanishing partial volumes of intersected grid cells are removed by a two-term asymptotic approach. In contrast to the 2D method presented by two of the authors in [M.~Oevermann, R.~Klein: A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J.~Comp.~Phys.~219 (2006)] we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz. This simplifies the treatment of the different cut-cell types and avoids additional special operations for degenerated interface topologies. The resulting set of linear equations has been solved with a BiCGSTAB solver preconditioned with an algebraic multigrid. In various testcases -- including large coefficient ratios and non-smooth interfaces -- the method achieves second order of accuracy in the L_inf and L_2 norm.}, language = {en} } @misc{OevermannKlein, author = {Oevermann, Michael and Klein, Rupert}, title = {A cartesian grid finite volume method for the solution of the Poisson equation with variable coefficients and embedded interfaces}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8996}, number = {06-05}, abstract = {We present a finite volume method for the solution of the two-dimensional Poisson equation \$ \nabla\cdot( \beta( {\mbox{\boldmath \$x\$}}) \nabla u({\mbox{\boldmath \$x\$}})) = f(\mbox{\boldmath \$x\$}) \$ with variable, discontinuous coefficients and solution discontinuities on irregular domains. The method uses bilinear ansatz functions on Cartesian grids for the solution \$u({\mbox{\boldmath \$x\$})\$ resulting in a compact nine-point stencil. The resulting linear problem has been solved with a standard multigrid solver. Singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two-step asymptotic approach. The method achieves second order of accuracy in the \$L^\infty\$ and \$L^2\$ norm.}, language = {en} } @misc{BraessDeuflhardLipnikov, author = {Braess, Dietrich and Deuflhard, Peter and Lipnikov, Konstantin}, title = {A Subspace Cascadic Multigrid Method for Mortar Elements}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3954}, number = {SC-99-07}, abstract = {A cascadic multigrid (CMG) method for elliptic problems with strong material jumps is proposed and analyzed. Non--matching grids at interfaces between subdomains are allowed and treated by mortar elements. The arising saddle point problems are solved by a subspace confined conjugate gradient method as smoother for the CMG. Details of algorithmic realization including adaptivity are elaborated. Numerical results illustrate the efficiency of this CMG algorithm.}, language = {en} } @misc{Deuflhard, author = {Deuflhard, Peter}, title = {Differential Equations in Technology and Medicine. Computational Concepts, Adaptive Algorithms, and Virtual Labs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4223}, number = {SC-99-34}, abstract = {This series of lectures has been given to a class of mathematics postdocs at a European summer school on Computational Mathematics Driven by Industrial Applications in Martina Franca, Italy (organized by CIME). It deals with a variety of challenging real life problems selected from clinical cancer therapy, communication technology, polymer production, and pharmaceutical drug design. All of these problems from rather diverse application areas share two common features: (a) they have been modelled by various differential equations -- elliptic, parabolic, or Schr{\"o}dinger--type partial differential equations, countable ordinary diffential equations, or Hamiltonian systems, (b) their numerical solution has turned out to be real challenge to computational mathematics.}, language = {en} }