TY - GEN A1 - Götschel, Sebastian A1 - Schiela, Anton A1 - Weiser, Martin T1 - Kaskade 7 -- a Flexible Finite Element Toolbox N2 - Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the Dune interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the mix-and-match approach to discretizing systems of PDEs with different ansatz and test spaces for all variables. We describe the mathematical concepts behind the library as well as its structure, illustrating its use at several examples on the way. T3 - ZIB-Report - 19-48 KW - finite elements KW - generic programming KW - partial differential equations Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-74616 SN - 1438-0064 ER - TY - GEN A1 - Lubkoll, Lars A1 - Schiela, Anton A1 - Weiser, Martin T1 - An affine covariant composite step method for optimization with PDEs as equality constraints N2 - We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. We discuss preconditioning strategies for the iterative solution of the arising linear systems with projected conjugate gradient. Numerical results are shown for optimal control problems subject to a nonlinear heat equation and subject to nonlinear elastic equations arising from an implant design problem in craniofacial surgery. T3 - ZIB-Report - 15-09 KW - composite step methods KW - cubic regularization KW - affine covariant KW - optimization with PDEs Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-53954 SN - 1438-0064 ER - TY - JOUR A1 - Lubkoll, Lars A1 - Schiela, Anton A1 - Weiser, Martin T1 - An affine covariant composite step method for optimization with PDEs as equality constraints JF - Optimization Methods and Software N2 - We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. We discuss preconditioning strategies for the iterative solution of the arising linear systems with projected conjugate gradient. Numerical results are shown for optimal control problems subject to a nonlinear heat equation and subject to nonlinear elastic equations arising from an implant design problem in craniofacial surgery. Y1 - 2017 U6 - https://doi.org/10.1080/10556788.2016.1241783 VL - 32 IS - 5 SP - 1132 EP - 1161 ER - TY - BOOK A1 - Weiser, Martin T1 - Inside Finite Elements N2 - All relevant implementation aspects of finite element methods are discussed in this book. The focus is on algorithms and data structures as well as on their concrete implementation. Theory is covered as far as it gives insight into the construction of algorithms.Throughout the exercises a complete FE-solver for scalar 2D problems will be implemented in Matlab/Octave. Y1 - 2016 PB - De Gruyter ER - TY - THES A1 - Moldenhauer, Marian T1 - Adaptive Algorithms in Optimization under PDE Constraints N2 - In this thesis, adaptive algorithms in optimization under PDE constraints have been inves- tigated. In its application, the aim of optimization is to increase the longevity of implants, namely the hip joint implant, and in doing so to minimize stress shielding and simultaneously minimize the influence of locally high stresses, that, above a threshold value, are malign to the bone structure. Under the constraint of the equilibrium of forces, describing an elastodynamic setup, coupled with a contact inequality condition, a computationally expensive problem formulation is given. The first step to make the solution of the given problem possible and efficient was to change over to the spatial equilibrium equation, thus rendering an elastostatic setup. Subsequently the intrinsically dynamic motions – trajectories in the load domain – were converted to the static setup. Thus, the trajectories are marginalized to the load domain and characterized with probability distributions. Therefore the solving of the PDE constraint, the contact problem, is simplified. Yet in the whole optimization process, the solving of the PDE, the spatial equilibrium equation together with the contact condition has the most expensive contribution still and hence needed further reduction. This was achieved by application of Kriging interpolation to the load responses of the integrated distribution of stress difference and the maximum stresses. The interpolation of the two response surfaces only needs comparatively few PDE solves to set up the models. Moreover, the Kriging models can be adaptively extended by sequentially adding sample-response pairs. For this the Kriging inherent variance is used to estimate ideal new sample locations with maximum variance values. In doing so, the overall interpolation variance and therefore the interpolation error is reduced. For the integration of the integrated stress differences and penalty values on the relative high dimensional load domain Monte Carlo integration was implemented, averting the curse of dimension. Here, the motion’s probability distribution combined with patient specific data of motion frequencies is taken advantage of, making obsolete the use of the otherwise necessary importance sampling. Throughout the optimization, the FE-discretization error and the subsequently attached errors entering the solution process via PDE discretization and approximative solving of the PDE, Kriging interpolation and Monte Carlo integration need to decrease. While the FE-discretization error and the solution of the elastostatic contact problem were assumed precise enough, numerics showed, that the interpolation and integration errors can be controlled by adaptive refinement of the respective methods. For this purpose comparable error quantities for the particular algorithms were introduced and effectively put to use. For the implant position’s optimization, the derivative of the objective function was derived using the implicit function theorem. As the FE-discretization changes with implant position modifications big enough, a special line search had to be used to deal with the discontinuities in the objective function. The interplay and performance of the subalgorithms was demonstrated numerically on a reduced 2D setup of a hip joint with and without the implant. Consequently the load domain and the control variable were also limited to the 2D case. Y1 - 2021 ER -