Journal of geometric analysis/The Journal of geometric analysis, Apr 29, 2024
Multi-material design optimization problems can, after discretization, be solved by the iterative... more Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman-Morrison-Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two-at a first sight-very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline stage. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.
Zenodo (CERN European Organization for Nuclear Research), Mar 22, 2023
The digital transformation and, thus, the use of new digital technologies not only has a substant... more The digital transformation and, thus, the use of new digital technologies not only has a substantial impact on society and companies but also on science. Analog documentation as we have known it for centuries will eventually be replaced by intelligent and FAIR (Findable, Accessible, Interoperable, and Reusable) systems. In addition to the actual research data and results, metadata now plays an important role, not only in the use of individual, independently existing projects, but also for future use by other scientists and in interdisciplinary research groups and disciplines. The solution presented here of an electronic laboratory notebook and laboratory information management system (ELN-LIMS) based on the openBIS (open Biology Information System) environment offers interesting features and advantages, especially for interdisciplinary work. The Collaborative Research Centre (CRC) 1411 "Design of Particulate Products" of the German Research Foundation is characterized by the cooperation of different working groups of synthesis, characterization, and simulation, and therefore serves as a model environment to present the implementation of openBIS. OpenBIS, as an open source ELN-LIMS solution following FAIR principles, provides a common set of general entries with the possibility of sharing and linking (meta-)data to improve the scientific exchange between all users.
In this contribution, we present a full overview of the continuous stochastic gradient (CSG) meth... more In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function requires some form of integration, e.g., expected values. Since approximating the integration by a fixed quadrature rule can introduce artificial local solutions into the problem while simultaneously raising the computational effort, stochastic optimization schemes have become increasingly popular in such contexts. However, known stochastic gradient type methods are typically limited to expected risk functions and inherently require many iterations. The latter is particularly problematic, if the evaluation of the cost function involves solving multiple state equations, given, e.g., in form of partial differential equations. To overcome these drawbacks, a recent article introduced the CSG method, which reuses old gradient sample information via the calculation of design dependent integration weights to obtain a better approximation to the full gradient. While in the original CSG paper convergence of a subsequence was established for a diminishing step size, here, we provide a complete convergence analysis of CSG for constant step sizes and an Armijotype line search. Moreover, new methods to obtain the integration 1
The material characterization revealed a homogeneous size distribution of the particles, while th... more The material characterization revealed a homogeneous size distribution of the particles, while the specific surface area of pure and doped ceria particles depends on the process conditions between 39 and 81 m 2 g-1. The unique emission spectrum of Eu 3+ demonstrates the efficient Eu doping of ceria NPs. The unique (optical) properties of these particles were characterized in time-resolved cryo luminescence experiments. The catalytic activity for the conversion of CO into CO 2 is successfully proven.
A coupled two-scale material optimization approach in linear elasticity using lattice structures ... more A coupled two-scale material optimization approach in linear elasticity using lattice structures is presented. In a previous work [1], a coupled two-scale material optimization problem using homogenized lattice structures is presented. In the current work, the homogenization as upscaling formula is replaced by a modified multiscale finite element method (MSFEM), based on [2].
We present a new approach and an algorithm for solving two-scale material optimization problems t... more We present a new approach and an algorithm for solving two-scale material optimization problems to optimize the behaviour of a fluid-saturated porous medium in a given domain. While the state problem is governed by the Biot model describing the fluid–structure interaction in homogenized poroelastic structures, the approach is widely applicable to multiphysics problems involving several macroscopic fields in which homogenization provides the relationship between the microconfigurations and the macroscopic mathematical model. The optimization variables describe the local microstructure design by virtue of the pore shape which determines the effective medium properties, namely the material coefficients, computed by the homogenization method. The numerical optimization strategy involves (a) precomputing a database of the material coefficients associated with the geometric parameters and (b) applying the sequential global programming (SGP) method for solving the problem of macroscopicall...
Structural and Multidisciplinary Optimization, May 31, 2020
This paper presents a novel method for the solution of a particular class of structural optimzati... more This paper presents a novel method for the solution of a particular class of structural optimzation problems: the continuous stochastic gradient method (CSG). In the simplest case, we assume that the objective function is given as an integral of a desired property over a continuous parameter set. The application of a quadrature rule for the approximation of the integral can give rise to artificial and undesired local minima. However, the CSG method does not rely on an approximation of the integral, instead utilizing gradient approximations from previous iterations in an optimal way. Although the CSG method does not require more than the solution of one state problem (of infinitely many) per optimization iteration, it is possible to prove in a mathematically rigorous way that the function value as well as the full gradient of the objective can be approximated with arbitrary precision in the course of the optimization process. Moreover, numerical experiments for a linear elastic problem with infinitely many load cases are described. For the chosen example, the CSG method proves to be clearly superior compared to the classic stochastic gradient (SG) and the stochastic average gradient (SAG) method.
In this article a topology optimization method is developed, which is aware of material uncertain... more In this article a topology optimization method is developed, which is aware of material uncertainties. The uncertainties are handled in a worstcase sense, i.e. the worst possible material distribution over a given uncertainty set is taken into account for each topology. The worst-case approach leads to a minimax problem, which is analyzed throughout the paper. A conservative convex relaxation for the inner maximization problem is suggested, which allows to treat the minimax problem by minimization of an optimal value function. A Tikhonov type and a barrier regularization scheme are developed, which render the resulting minimization problem continuously differentiable. The barrier regularization scheme turns out to be more suitable for the practical solution of the problem, as it can be closely linked to a highly efficient interior point approach used for the evaluation of the optimal value function and its gradient. Based on this, the outer minimization problem can be approached by a gradient based optimization solver like the method of moving asymptotes. Examples from additive manufacturing as well as material degradation are examined, demonstrating the practical applicability as well as the efficiency of the suggested method. Finally, the impact of the convex relaxation of the inner problem is investigated. It is observed that for large material deviations, the robust solution may become too conservative. As a remedy, a RAMP-type continuation scheme for the inner problem is suggested and numerically tested.
Currently, few approaches are available for mixed-integer nonlinear robust optimization. Those th... more Currently, few approaches are available for mixed-integer nonlinear robust optimization. Those that do exist typically either require restrictive assumptions on the problem structure or do not guarantee robust protection. In this work, we develop an algorithm for convex mixed-integer nonlinear robust optimization problems where a key feature is that the method does not rely on a specific structure of the inner worst-case (adversarial) problem and allows the latter to be non-convex. A major challenge of such a general nonlinear setting is ensuring robust protection, as this calls for a global solution of the non-convex adversarial problem. Our method is able to achieve this up to a tolerance, by requiring worst-case evaluations only up to a certain precision. For example, the necessary assumptions can be met by approximating a non-convex adversarial via piecewise relaxations and solving the resulting problem up to any requested error as a mixed-integer linear problem.In our approach,...
Communications in Nonlinear Science and Numerical Simulation, 2019
We present a mathematical approach to optimize the material distribution for fluid transport in u... more We present a mathematical approach to optimize the material distribution for fluid transport in unsaturated porous media. Our benchmark problem is a simplified diaper model as an exemplary liquid absorber. Our model has up to three materials with vastly different properties, which in the reference configuration are arranged in parallel layers. Here, swelling is neglected and the geometry of a swollen diaper is assumed and treated as a porous medium of high porosity. The imbibition process is then simulated by solving an unsaturated flow problem based on Richards' equation. Our aim is to maximize the amount of absorbed liquid by redistributing the materials. To this end, a density based multi-material topology optimization (based on the well known SIMP model) is performed. A sensitivity analysis for the nonlinear transient problem is provided, which enables the application of first order optimization solvers. We perform two-and three-material optimization and discuss several variants of the problem setting. We present designs with up to 45% more absorbed liquid compared to the reference configuration.
In this paper we study feasibility and infeasibility of nonlinear two-stage fully adjustable robu... more In this paper we study feasibility and infeasibility of nonlinear two-stage fully adjustable robust feasibility problems with an empty first stage. This is equivalent to deciding whether the uncertainty set is contained within the projection of the feasible region onto the uncertaintyspace. Moreover, the considered sets are assumed to be described by polynomials. For answering this question, two very general approaches using methods from polynomial optimization are presented-one for showing feasibility and one for showing infeasibility. The developed methods are approximated through sum of squares polynomials and solved using semidefinite programs. Deciding robust feasibility and infeasibility is important for gas network operations, which is a nonconvex feasibility problem where the feasible set is described by a composition of polynomials with the absolute value function. Concerning the gas network problem, different topologies are considered. It is shown that a tree structured network can be decided exactly using linear programming. Furthermore, a method is presented to reduce a tree network with one additional arc to a single cycle network. In this case, the problem can be decided by eliminating the absolute value functions and solving the resulting linearly many polynomial optimization problems. Lastly, the effectivity of the methods is tested on a variety of small cyclic networks. It turns out that for instances where robust feasibility or infeasibility can be decided successfully, level 2 or level 3 of the Lasserre relaxation hierarchy typically is sufficient.
We present an algorithm for the solution of static output feedback problems formulated as semidef... more We present an algorithm for the solution of static output feedback problems formulated as semidefinite programs with bilinear matrix inequality constraints and collected in the library COMPl e ib. The algorithm, based on the generalized augmented Lagrangian technique, is implemented in the publicly available general purpose software PENBMI. Numerical results demonstrate the behavior of the code.
Proceedings of the 44th IEEE Conference on Decision and Control
An algebraic formulation is proposed for the static output feedback (SOF) problem: the Hermite st... more An algebraic formulation is proposed for the static output feedback (SOF) problem: the Hermite stability criterion is applied on the closed-loop characteristic polynomial, resulting in a non-convex bilinear matrix inequality (BMI) optimization problem for SIMO or MISO systems. As a result, the BMI problem is formulated directly in the controller parameters, without additional Lyapunov variables. The publicly available solver PENBMI 2.0 interfaced with YALMIP 3.0 is then applied to solve benchmark examples. Implementation and numerical aspects are widely discussed.
A recent article introduced the continuous stochastic gradient method (CSG) for the efficient sol... more A recent article introduced the continuous stochastic gradient method (CSG) for the efficient solution of a class of stochastic optimization problems. While the applicability of known stochastic gradient type methods is typically limited to expected risk functions, no such limitation exists for CSG. This advantage stems from the computation of design dependent integration weights, allowing for optimal usage of available information and therefore stronger convergence properties. However, the nature of the formula used for these integration weights essentially limited the practical applicability of this method to problems in which stochasticity enters via a low-dimensional and sufficiently simple probability distribution. In this paper we significantly extend the scope of the CSG method by presenting alternative ways to calculate the integration weights. A full convergence analysis for this new variant of the CSG method is presented and its efficiency is demonstrated in comparison to more classical stochastic gradient methods by means of a number of problem classes relevant to stochastic optimization and machine learning.
Structural and Multidisciplinary Optimization, Apr 13, 2023
We present a feature-mapping topology optimization approach, in which curved features are paramet... more We present a feature-mapping topology optimization approach, in which curved features are parametrized as piecewise linear splines smoothly rounded by arcs. The motivation for our contribution to the tool set of feature-mapping methods is the optimization of structures manufactured by variable angle continuous fiber-reinforced filaments. For this reason, the feature's geometry should be able to represent long, curved fiber objects satisfying manufacturing constraints, such as minimum turning radius. The proposed model has been chosen with special care for rigorous continuous differentiability, as well as an efficient analytical evaluation of the signed distance field to the spline. The geometrical description and sensitivity analysis of the spline model are developed fully analytically and then mapped to a discretized pseudo-density field for finite element analysis. For the fiber-reinforced material formulation, we also present a new combine step for individual features, in which the best possible angle for the combined features is searched. The model and results are presented in a two-dimensional setting.
Structural and Multidisciplinary Optimization, Feb 24, 2020
In this article a topology optimization method is developed, which is aware of material uncertain... more In this article a topology optimization method is developed, which is aware of material uncertainties. The uncertainties are handled in a worstcase sense, i.e. the worst possible material distribution over a given uncertainty set is taken into account for each topology. The worst-case approach leads to a minimax problem, which is analyzed throughout the paper. A conservative convex relaxation for the inner maximization problem is suggested, which allows to treat the minimax problem by minimization of an optimal value function. A Tikhonov type and a barrier regularization scheme are developed, which render the resulting minimization problem continuously differentiable. The barrier regularization scheme turns out to be more suitable for the practical solution of the problem, as it can be closely linked to a highly efficient interior point approach used for the evaluation of the optimal value function and its gradient. Based on this, the outer minimization problem can be approached by a gradient based optimization solver like the method of moving asymptotes. Examples from additive manufacturing as well as material degradation are examined, demonstrating the practical applicability as well as the efficiency of the suggested method. Finally, the impact of the convex relaxation of the inner problem is investigated. It is observed that for large material deviations, the robust solution may become too conservative. As a remedy, a RAMP-type continuation scheme for the inner problem is suggested and numerically tested.
Structural and Multidisciplinary Optimization, May 20, 2016
We consider the problem of parametric material and simultaneous topology optimization of an elast... more We consider the problem of parametric material and simultaneous topology optimization of an elastic continuum. To ensure existence of solutions to the proposed optimization problem and to enable the imposition of a deliberate maximal material grading, two approaches are adopted and combined. The first imposes pointwise bounds on design variable gradients, whilst the second applies a filtering technique based on a convolution product. For the topology optimization, the parametrized material is multiplied with a penalized continuous density variable. We suggest a finite element discretization of the problem and provide a proof of convergence for the finite element solutions to solutions of the continuous problem. The convergence proof also implies the absence of checkerboards. The concepts are demonstrated by means of numerical examples using a number of different material parametrizations and comparing the results to global lower bounds.
PENLAB is an open source software package for nonlinear optimization, linear and nonlinear semide... more PENLAB is an open source software package for nonlinear optimization, linear and nonlinear semidefinite optimization and any combination of these. It is written entirely in MATLAB. PENLAB is a young brother of our code PENNON [23] and of a new implementation from NAG [1]: it can solve the same classes of problems and uses the same algorithm. Unlike PENNON, PENLAB is open source and allows the user not only to solve problems but to modify various parts of the algorithm. As such, PENLAB is particularly suitable for teaching and research purposes and for testing new algorithmic ideas. In this article, after a brief presentation of the underlying algorithm, we focus on practical use of the solver, both for general problem classes and for specific practical problems.
Journal of geometric analysis/The Journal of geometric analysis, Apr 29, 2024
Multi-material design optimization problems can, after discretization, be solved by the iterative... more Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman-Morrison-Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two-at a first sight-very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline stage. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.
Zenodo (CERN European Organization for Nuclear Research), Mar 22, 2023
The digital transformation and, thus, the use of new digital technologies not only has a substant... more The digital transformation and, thus, the use of new digital technologies not only has a substantial impact on society and companies but also on science. Analog documentation as we have known it for centuries will eventually be replaced by intelligent and FAIR (Findable, Accessible, Interoperable, and Reusable) systems. In addition to the actual research data and results, metadata now plays an important role, not only in the use of individual, independently existing projects, but also for future use by other scientists and in interdisciplinary research groups and disciplines. The solution presented here of an electronic laboratory notebook and laboratory information management system (ELN-LIMS) based on the openBIS (open Biology Information System) environment offers interesting features and advantages, especially for interdisciplinary work. The Collaborative Research Centre (CRC) 1411 "Design of Particulate Products" of the German Research Foundation is characterized by the cooperation of different working groups of synthesis, characterization, and simulation, and therefore serves as a model environment to present the implementation of openBIS. OpenBIS, as an open source ELN-LIMS solution following FAIR principles, provides a common set of general entries with the possibility of sharing and linking (meta-)data to improve the scientific exchange between all users.
In this contribution, we present a full overview of the continuous stochastic gradient (CSG) meth... more In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function requires some form of integration, e.g., expected values. Since approximating the integration by a fixed quadrature rule can introduce artificial local solutions into the problem while simultaneously raising the computational effort, stochastic optimization schemes have become increasingly popular in such contexts. However, known stochastic gradient type methods are typically limited to expected risk functions and inherently require many iterations. The latter is particularly problematic, if the evaluation of the cost function involves solving multiple state equations, given, e.g., in form of partial differential equations. To overcome these drawbacks, a recent article introduced the CSG method, which reuses old gradient sample information via the calculation of design dependent integration weights to obtain a better approximation to the full gradient. While in the original CSG paper convergence of a subsequence was established for a diminishing step size, here, we provide a complete convergence analysis of CSG for constant step sizes and an Armijotype line search. Moreover, new methods to obtain the integration 1
The material characterization revealed a homogeneous size distribution of the particles, while th... more The material characterization revealed a homogeneous size distribution of the particles, while the specific surface area of pure and doped ceria particles depends on the process conditions between 39 and 81 m 2 g-1. The unique emission spectrum of Eu 3+ demonstrates the efficient Eu doping of ceria NPs. The unique (optical) properties of these particles were characterized in time-resolved cryo luminescence experiments. The catalytic activity for the conversion of CO into CO 2 is successfully proven.
A coupled two-scale material optimization approach in linear elasticity using lattice structures ... more A coupled two-scale material optimization approach in linear elasticity using lattice structures is presented. In a previous work [1], a coupled two-scale material optimization problem using homogenized lattice structures is presented. In the current work, the homogenization as upscaling formula is replaced by a modified multiscale finite element method (MSFEM), based on [2].
We present a new approach and an algorithm for solving two-scale material optimization problems t... more We present a new approach and an algorithm for solving two-scale material optimization problems to optimize the behaviour of a fluid-saturated porous medium in a given domain. While the state problem is governed by the Biot model describing the fluid–structure interaction in homogenized poroelastic structures, the approach is widely applicable to multiphysics problems involving several macroscopic fields in which homogenization provides the relationship between the microconfigurations and the macroscopic mathematical model. The optimization variables describe the local microstructure design by virtue of the pore shape which determines the effective medium properties, namely the material coefficients, computed by the homogenization method. The numerical optimization strategy involves (a) precomputing a database of the material coefficients associated with the geometric parameters and (b) applying the sequential global programming (SGP) method for solving the problem of macroscopicall...
Structural and Multidisciplinary Optimization, May 31, 2020
This paper presents a novel method for the solution of a particular class of structural optimzati... more This paper presents a novel method for the solution of a particular class of structural optimzation problems: the continuous stochastic gradient method (CSG). In the simplest case, we assume that the objective function is given as an integral of a desired property over a continuous parameter set. The application of a quadrature rule for the approximation of the integral can give rise to artificial and undesired local minima. However, the CSG method does not rely on an approximation of the integral, instead utilizing gradient approximations from previous iterations in an optimal way. Although the CSG method does not require more than the solution of one state problem (of infinitely many) per optimization iteration, it is possible to prove in a mathematically rigorous way that the function value as well as the full gradient of the objective can be approximated with arbitrary precision in the course of the optimization process. Moreover, numerical experiments for a linear elastic problem with infinitely many load cases are described. For the chosen example, the CSG method proves to be clearly superior compared to the classic stochastic gradient (SG) and the stochastic average gradient (SAG) method.
In this article a topology optimization method is developed, which is aware of material uncertain... more In this article a topology optimization method is developed, which is aware of material uncertainties. The uncertainties are handled in a worstcase sense, i.e. the worst possible material distribution over a given uncertainty set is taken into account for each topology. The worst-case approach leads to a minimax problem, which is analyzed throughout the paper. A conservative convex relaxation for the inner maximization problem is suggested, which allows to treat the minimax problem by minimization of an optimal value function. A Tikhonov type and a barrier regularization scheme are developed, which render the resulting minimization problem continuously differentiable. The barrier regularization scheme turns out to be more suitable for the practical solution of the problem, as it can be closely linked to a highly efficient interior point approach used for the evaluation of the optimal value function and its gradient. Based on this, the outer minimization problem can be approached by a gradient based optimization solver like the method of moving asymptotes. Examples from additive manufacturing as well as material degradation are examined, demonstrating the practical applicability as well as the efficiency of the suggested method. Finally, the impact of the convex relaxation of the inner problem is investigated. It is observed that for large material deviations, the robust solution may become too conservative. As a remedy, a RAMP-type continuation scheme for the inner problem is suggested and numerically tested.
Currently, few approaches are available for mixed-integer nonlinear robust optimization. Those th... more Currently, few approaches are available for mixed-integer nonlinear robust optimization. Those that do exist typically either require restrictive assumptions on the problem structure or do not guarantee robust protection. In this work, we develop an algorithm for convex mixed-integer nonlinear robust optimization problems where a key feature is that the method does not rely on a specific structure of the inner worst-case (adversarial) problem and allows the latter to be non-convex. A major challenge of such a general nonlinear setting is ensuring robust protection, as this calls for a global solution of the non-convex adversarial problem. Our method is able to achieve this up to a tolerance, by requiring worst-case evaluations only up to a certain precision. For example, the necessary assumptions can be met by approximating a non-convex adversarial via piecewise relaxations and solving the resulting problem up to any requested error as a mixed-integer linear problem.In our approach,...
Communications in Nonlinear Science and Numerical Simulation, 2019
We present a mathematical approach to optimize the material distribution for fluid transport in u... more We present a mathematical approach to optimize the material distribution for fluid transport in unsaturated porous media. Our benchmark problem is a simplified diaper model as an exemplary liquid absorber. Our model has up to three materials with vastly different properties, which in the reference configuration are arranged in parallel layers. Here, swelling is neglected and the geometry of a swollen diaper is assumed and treated as a porous medium of high porosity. The imbibition process is then simulated by solving an unsaturated flow problem based on Richards' equation. Our aim is to maximize the amount of absorbed liquid by redistributing the materials. To this end, a density based multi-material topology optimization (based on the well known SIMP model) is performed. A sensitivity analysis for the nonlinear transient problem is provided, which enables the application of first order optimization solvers. We perform two-and three-material optimization and discuss several variants of the problem setting. We present designs with up to 45% more absorbed liquid compared to the reference configuration.
In this paper we study feasibility and infeasibility of nonlinear two-stage fully adjustable robu... more In this paper we study feasibility and infeasibility of nonlinear two-stage fully adjustable robust feasibility problems with an empty first stage. This is equivalent to deciding whether the uncertainty set is contained within the projection of the feasible region onto the uncertaintyspace. Moreover, the considered sets are assumed to be described by polynomials. For answering this question, two very general approaches using methods from polynomial optimization are presented-one for showing feasibility and one for showing infeasibility. The developed methods are approximated through sum of squares polynomials and solved using semidefinite programs. Deciding robust feasibility and infeasibility is important for gas network operations, which is a nonconvex feasibility problem where the feasible set is described by a composition of polynomials with the absolute value function. Concerning the gas network problem, different topologies are considered. It is shown that a tree structured network can be decided exactly using linear programming. Furthermore, a method is presented to reduce a tree network with one additional arc to a single cycle network. In this case, the problem can be decided by eliminating the absolute value functions and solving the resulting linearly many polynomial optimization problems. Lastly, the effectivity of the methods is tested on a variety of small cyclic networks. It turns out that for instances where robust feasibility or infeasibility can be decided successfully, level 2 or level 3 of the Lasserre relaxation hierarchy typically is sufficient.
We present an algorithm for the solution of static output feedback problems formulated as semidef... more We present an algorithm for the solution of static output feedback problems formulated as semidefinite programs with bilinear matrix inequality constraints and collected in the library COMPl e ib. The algorithm, based on the generalized augmented Lagrangian technique, is implemented in the publicly available general purpose software PENBMI. Numerical results demonstrate the behavior of the code.
Proceedings of the 44th IEEE Conference on Decision and Control
An algebraic formulation is proposed for the static output feedback (SOF) problem: the Hermite st... more An algebraic formulation is proposed for the static output feedback (SOF) problem: the Hermite stability criterion is applied on the closed-loop characteristic polynomial, resulting in a non-convex bilinear matrix inequality (BMI) optimization problem for SIMO or MISO systems. As a result, the BMI problem is formulated directly in the controller parameters, without additional Lyapunov variables. The publicly available solver PENBMI 2.0 interfaced with YALMIP 3.0 is then applied to solve benchmark examples. Implementation and numerical aspects are widely discussed.
A recent article introduced the continuous stochastic gradient method (CSG) for the efficient sol... more A recent article introduced the continuous stochastic gradient method (CSG) for the efficient solution of a class of stochastic optimization problems. While the applicability of known stochastic gradient type methods is typically limited to expected risk functions, no such limitation exists for CSG. This advantage stems from the computation of design dependent integration weights, allowing for optimal usage of available information and therefore stronger convergence properties. However, the nature of the formula used for these integration weights essentially limited the practical applicability of this method to problems in which stochasticity enters via a low-dimensional and sufficiently simple probability distribution. In this paper we significantly extend the scope of the CSG method by presenting alternative ways to calculate the integration weights. A full convergence analysis for this new variant of the CSG method is presented and its efficiency is demonstrated in comparison to more classical stochastic gradient methods by means of a number of problem classes relevant to stochastic optimization and machine learning.
Structural and Multidisciplinary Optimization, Apr 13, 2023
We present a feature-mapping topology optimization approach, in which curved features are paramet... more We present a feature-mapping topology optimization approach, in which curved features are parametrized as piecewise linear splines smoothly rounded by arcs. The motivation for our contribution to the tool set of feature-mapping methods is the optimization of structures manufactured by variable angle continuous fiber-reinforced filaments. For this reason, the feature's geometry should be able to represent long, curved fiber objects satisfying manufacturing constraints, such as minimum turning radius. The proposed model has been chosen with special care for rigorous continuous differentiability, as well as an efficient analytical evaluation of the signed distance field to the spline. The geometrical description and sensitivity analysis of the spline model are developed fully analytically and then mapped to a discretized pseudo-density field for finite element analysis. For the fiber-reinforced material formulation, we also present a new combine step for individual features, in which the best possible angle for the combined features is searched. The model and results are presented in a two-dimensional setting.
Structural and Multidisciplinary Optimization, Feb 24, 2020
In this article a topology optimization method is developed, which is aware of material uncertain... more In this article a topology optimization method is developed, which is aware of material uncertainties. The uncertainties are handled in a worstcase sense, i.e. the worst possible material distribution over a given uncertainty set is taken into account for each topology. The worst-case approach leads to a minimax problem, which is analyzed throughout the paper. A conservative convex relaxation for the inner maximization problem is suggested, which allows to treat the minimax problem by minimization of an optimal value function. A Tikhonov type and a barrier regularization scheme are developed, which render the resulting minimization problem continuously differentiable. The barrier regularization scheme turns out to be more suitable for the practical solution of the problem, as it can be closely linked to a highly efficient interior point approach used for the evaluation of the optimal value function and its gradient. Based on this, the outer minimization problem can be approached by a gradient based optimization solver like the method of moving asymptotes. Examples from additive manufacturing as well as material degradation are examined, demonstrating the practical applicability as well as the efficiency of the suggested method. Finally, the impact of the convex relaxation of the inner problem is investigated. It is observed that for large material deviations, the robust solution may become too conservative. As a remedy, a RAMP-type continuation scheme for the inner problem is suggested and numerically tested.
Structural and Multidisciplinary Optimization, May 20, 2016
We consider the problem of parametric material and simultaneous topology optimization of an elast... more We consider the problem of parametric material and simultaneous topology optimization of an elastic continuum. To ensure existence of solutions to the proposed optimization problem and to enable the imposition of a deliberate maximal material grading, two approaches are adopted and combined. The first imposes pointwise bounds on design variable gradients, whilst the second applies a filtering technique based on a convolution product. For the topology optimization, the parametrized material is multiplied with a penalized continuous density variable. We suggest a finite element discretization of the problem and provide a proof of convergence for the finite element solutions to solutions of the continuous problem. The convergence proof also implies the absence of checkerboards. The concepts are demonstrated by means of numerical examples using a number of different material parametrizations and comparing the results to global lower bounds.
PENLAB is an open source software package for nonlinear optimization, linear and nonlinear semide... more PENLAB is an open source software package for nonlinear optimization, linear and nonlinear semidefinite optimization and any combination of these. It is written entirely in MATLAB. PENLAB is a young brother of our code PENNON [23] and of a new implementation from NAG [1]: it can solve the same classes of problems and uses the same algorithm. Unlike PENNON, PENLAB is open source and allows the user not only to solve problems but to modify various parts of the algorithm. As such, PENLAB is particularly suitable for teaching and research purposes and for testing new algorithmic ideas. In this article, after a brief presentation of the underlying algorithm, we focus on practical use of the solver, both for general problem classes and for specific practical problems.
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Papers by Michael Stingl