TY - GEN A1 - Beck, Rudolf A1 - Erdmann, Bodo A1 - Roitzsch, Rainer T1 - KASKADE 3.0 - An Object Oriented Adaptive Finite Element Code N2 - KASKADE 3.0 was developed for the solution of partial differential equations in one, two, or three space dimensions. Its object-oriented implementation concept is based on the programming language C++$\,$.~Adaptive finite element techniques are employed to provide solution procedures of optimal computational complexity. This implies a posteriori error estimation, local mesh refinement and multilevel preconditioning. The program was designed both as a platform for further developments of adaptive multilevel codes and as a tool to tackle practical problems. Up to now we have implemented scalar problem types like stationary or transient heat conduction. The latter one is solved with the Rothe method, enabling adaptivity both in space and time. Some nonlinear phenomena like obstacle problems or two-phase Stefan problems are incorporated as well. Extensions to vector-valued functions and complex arithmetic are provided. %Such free boundary problems ... We have implemented several iterative solvers for both symmetric and unsymmetric systems together with multiplicative and additive multilevel preconditioners. Systems arising from the nonlinear problems can be solved with lately developed monotone multigrid methods. %An object-oriented concept was chosen for KASKADE~3.0, based on the programming %language C++$\,$. This should provide the desired extensibilty and clearly %reflect the structure of the code. %A direct sparse matrix solver (Harwell MA28) is included. T3 - ZIB-Report - TR-95-04 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5199 ER - TY - GEN A1 - Beck, Rudolf T1 - Algebraic Multigrid by Component Splitting for Edge Elements on Simplicial Triangulations N2 - Our focus is on Maxwell's equations in the low frequency range; two specific applications we aim at are time-stepping schemes for eddy current computations and the stationary double-curl equation for time-harmonic fields. We assume that the computational domain is discretized by triangles or tetrahedrons; for the finite element approximation we choose N\'{e}d\'{e}lec's $H(curl)$-conforming edge elements of the lowest order. For the solution of the arising linear equation systems we devise an algebraic multigrid preconditioner based on a spatial component splitting of the field. Mesh coarsening takes place in an auxiliary subspace, which is constructed with the aid of a nodal vector basis. Within this subspace coarse grids are created by exploiting the matrix graphs. Additionally, we have to cope with the kernel of the $curl$-operator, which comprises a considerable part of the spectral modes on the grid. Fortunately, the kernel modes are accessible via a discrete Helmholtz decomposition of the fields; they are smoothed by additional algebraic multigrid cycles. Numerical experiments are included in order to assess the efficacy of the proposed algorithms. T3 - ZIB-Report - SC-99-40 KW - Algebraic multigrid KW - mesh coarsening KW - edge elements KW - N\'{e}d\'{e}lec spaces KW - Maxwell's equations Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4290 ER - TY - CHAP A1 - Beck, Rudolf A1 - Deuflhard, Peter A1 - Hege, Hans-Christian A1 - Seebass, Martin A1 - Stalling, Detlev ED - Hege, Hans-Christian ED - Polthier, Konrad T1 - Numerical Algorithms and Visualization in Medical Treament Planning T2 - Visualization and Mathematics Y1 - 1997 SP - 303 EP - 328 PB - Springer-Verlag ER - TY - JOUR A1 - Beck, Rudolf A1 - Deuflhard, Peter A1 - Hiptmair, Ralf A1 - Hoppe, Ronald H. W. A1 - Wohlmuth, Barbara T1 - Adaptive multilevel methods for edge element discretizations of Maxwell’s equations JF - Surv. Math. Ind. Y1 - 1999 VL - 8 IS - 3-4 SP - 271 EP - 312 ER - TY - CHAP A1 - Beck, Rudolf A1 - Deuflhard, Peter A1 - Wust, Peter A1 - Felix, Roland T1 - Electromagnetic Field Calculations by Finite Elements T2 - Proc. 16th Ann. Meeting Europ. Soc. Hyperthermic Oncology (ESHO-97), Berlin, Germany, April 1997 Y1 - 1997 SP - 77 ER - TY - CHAP A1 - Beck, Rudolf A1 - Hege, Hans-Christian A1 - Seebass, Martin A1 - Wust, Peter A1 - Deuflhard, Peter A1 - Felix, Roland ED - Franconi, Cafiero T1 - Adaptive finite element codes for numerical calculations in hyperthermia treatment planning T2 - Proc. of the 7th Int. Congress on Hyperthermic Oncology, Rome, Italy, April 1996 Y1 - 1996 VL - 2 SP - 515 EP - 517 ER - TY - CHAP A1 - Beck, Rudolf A1 - Erdmann, Bodo A1 - Roitzsch, Rainer ED - Arge, Erlend ED - Bruaset, Are Magnus ED - Langtangen, Hans Petter T1 - An Object-Oriented Adaptive Finite Element Code T2 - Modern Software Tools for Scientific Computing Y1 - 1997 SP - 105 EP - 124 PB - Birkhäuser, Boston ER - TY - CHAP A1 - Deuflhard, Peter A1 - Seebass, Martin A1 - Stalling, Detlev A1 - Beck, Rudolf A1 - Hege, Hans-Christian ED - Sydow, Achim T1 - Hyperthermia Treatment Planning in Clinical Cancer Therapy:Modelling, Simulation, and Visualization T2 - Computational Physics, Chemistry and Biology Y1 - 1997 VL - 3 PB - Wissenschaft und Technik Verlag CY - Plenary keynote talk, 15th IMACS World Congress 1997 on Scientific Computation, Modelling and Applied Mathematics ER - TY - GEN A1 - Beck, Rudolf T1 - Graph-Based Algebraic Multigrid for Lagrange-Type Finite Elements on Simplicial Meshes N2 - We present an algebraic multigrid preconditioner which uses only the graphs of system matrices. Some elementary coarsening rules are stated, from which an advancing front algorithm for the selection of coarse grid nodes is derived. This technique can be applied to linear Lagrange-type finite element discretizations; for higher-order elements an extension of the multigrid algorithm is provided. Both two- and three-dimensional second order elliptic problems can be handled. Numerical experiments show that the resulting convergence acceleration is comparable to classical geometric multigrid. T3 - ZIB-Report - SC-99-22 KW - Algebraic multigrid KW - mesh coarsening KW - preconditioning Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4100 ER - TY - GEN A1 - Seebass, Martin A1 - Beck, Rudolf A1 - Gellermann, Johanna A1 - Nadobny, Jacek A1 - Wust, Peter T1 - Electromagnetic phased arrays for regional hyperthermia -- optimal frequency and antenna arrangement N2 - 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. T3 - ZIB-Report - 00-28 KW - Pelvic heating KW - phased array optimization Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5961 ER -