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A nonlinear elastic shape averaging approach
 SIAM Journal on Imaging Sciences
, 2008
"... Abstract. A physically motivated approach is presented to compute a shape average of a given number of shapes. An elastic deformation is assigned to each shape. The shape average is then described as the common image under all elastic deformations of the given shapes, which minimizes the total elast ..."
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Abstract. A physically motivated approach is presented to compute a shape average of a given number of shapes. An elastic deformation is assigned to each shape. The shape average is then described as the common image under all elastic deformations of the given shapes, which minimizes the total elastic energy stored in these deformations. The underlying nonlinear elastic energy measures the local change of length, area, and volume. It is invariant under rigid body motions, and isometries are local minimizers. The model is relaxed involving a further energy which measures how well the elastic deformation image of a particular shape matches the average shape, and a suitable shape prior can be considered for the shape average. Shapes are represented via their edge sets, which also allows for an application to averaging image morphologies described via ensembles of edge sets. To make the approach computationally tractable, sharp edges are approximated via phase fields, and a corresponding variational phase field model is derived. Finite elements are applied for the spatial discretization, and a multiscale alternating minimization approach allows the efficient computation of shape averages in 2D and 3D. Various applications, e. g. averaging the shape of feet or human organs, underline the qualitative properties of the presented approach.
Algebraic Multigrid for Discrete Elliptic Second Order Problems. February 1996
, 1997
"... A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element ..."
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A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element
unknown title
, 2010
"... This thesis is protected by copyright which belongs to the author. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal perm ..."
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This thesis is protected by copyright which belongs to the author. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Please visit Sussex Research Online for more information and further details Primaldual active set methods for AllenCahn variational inequalities
MINIMUM WEIGHT TOPOLOGY OPTIMIZATION SUBJECT TO UNSTEADY HEAT EQUATION AND SPACETIME POINTWISE CONSTRAINTS – TOWARD AUTOMATIC OPTIMAL RISER DESIGN IN THE SHAPE CASTING PROCESS
"... Abstract. The automatic optimal design of feeding system in the shape casting process is considered, i.e., to find the optimal position, size, shape and topology of risers, and risernecks. It is formulated as a minimum weight topology optimization problem subjected to a nonlinear transient PDE and ..."
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Abstract. The automatic optimal design of feeding system in the shape casting process is considered, i.e., to find the optimal position, size, shape and topology of risers, and risernecks. It is formulated as a minimum weight topology optimization problem subjected to a nonlinear transient PDE and an infinite number of spacetime pointwise constraints. In addition to regularization and relaxation of the original model, an elegant bilevel reformulation of the optimization problem is introduced which makes it possible to manage the infinite number of design parameters and stateconstraints efficiently. The computational cost of this method is asymptotically independent from the number of design parameters and constraints. The validity and efficiency of the presented method are supported by several examples, from simple benchmarks to complex industrial castings. According to our numerical results, the presented approach makes a relatively complete solution to the problem of automatic optimal rider design in the shape casting process.
unknown title
"... no paig te a iza iza wi g’ ’ i liter issib al to y op inte 2012 Elsevier B.V. All rights reserved. to the e, for e mini, favor hibit p y form Penali tions r ue to proach continues as evidenced by the growing number of publications on this issue [10,39,25,26,51,43,27,50]. We limit the following lit ..."
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no paig te a iza iza wi g’ ’ i liter issib al to y op inte 2012 Elsevier B.V. All rights reserved. to the e, for e mini, favor hibit p y form Penali tions r ue to proach continues as evidenced by the growing number of publications on this issue [10,39,25,26,51,43,27,50]. We limit the following literature survey to density formulations but, as discussed above, the difficulty stems from a fundamental property of the original topology optimization problem and therefore similar measures are needed for other parameterizations of geometries, most notably the implicit functions methods (e.g., leMoreover the level of complexity of the final solutions (in fact all the admissible densities) is controlled directly by the regularity of the filtering kernel. Herein lies the major drawback: the smoother the filtering kernel, the larger the amount of the intermediate densities since the transition between the extreme values of density over the domain cannot occur too rapidly. Therefore, with more complexity control comes more ‘‘gray’ ’ regions and this, in some respect, undermines the basic premise of the density approach in that near characteristic functions are no longer recovered in the optimal regime (i.e., ‘‘0–1’ ’ or ‘‘blackandwhite’ ’ designs are not obtained).
Available online at www.sciencedirect.com
, 2014
"... www.elsevier.com/locate/cma A closer look at consistent operator splitting and its extensions for topology optimization ..."
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www.elsevier.com/locate/cma A closer look at consistent operator splitting and its extensions for topology optimization
nonlocal
, 2009
"... Primaldual active set methods for AllenCahn variational inequalities with ..."
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Primaldual active set methods for AllenCahn variational inequalities with
Mathematik
, 2009
"... Primaldual active set methods for AllenCahn variational inequalities with nonlocal constraints ..."
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Primaldual active set methods for AllenCahn variational inequalities with nonlocal constraints
An Optimal Solver for a KKTSystem arising from an InteriorPoint Formulation of a Topology Optimization Problem
, 2006
"... There are basically two approaches to solve optimal design problems with a partial differential equation, usually called the state equation, as a constraint. The usual procedure is to eliminate the state variables and the state equation and only optimize in the design space. Another possibility is ..."
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There are basically two approaches to solve optimal design problems with a partial differential equation, usually called the state equation, as a constraint. The usual procedure is to eliminate the state variables and the state equation and only optimize in the design space. Another possibility is to keep the state equation and to treat it as a constraint throughout the optimization progress. This formulation is called simultaneous or oneshot optimization. Then, in order to satisfy the optimality conditions, large scale indefinite linear systems (KKTsystems) have to be solved. This is the drawback, or better the challenge, of this alternative approach. If it is possible to construct an optimal solver to this KKTsystems, we can benefit from the expected speedup of the oneshot formulation. In this work we consider a multigrid based solver to such a KKTsystem, resulting from stress constrained topology optimization. As a proper smoothing procedure we use a multiplicative Schwarztype smoother. Here, in each iteration step of the smoother, several small local saddlepoint problems are solved. The numerical test examples show the typical multigrid convergence behaviour, i.e. asymptotic constant number of iterations and convergence rates.