Computer Methods in Applied Mechanics and Engineering, 2015
This paper applies topology optimisation to the design of structures with periodic microstructura... more This paper applies topology optimisation to the design of structures with periodic microstructural details without length scale separation, i.e. considering the complete macroscopic structure and its response, while resolving all microstructural details, as compared to the often used homogenisation approach. The approach takes boundary conditions into account and ensures connected and macroscopically optimised microstructures regardless of the difference in micro-and macroscopic length scales. This results in microstructures tailored for specific applications rather than specific properties.
Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures, 2014
A probabilistic approach is presented to account for geometric imperfections in topology optimiza... more A probabilistic approach is presented to account for geometric imperfections in topology optimization in order to ensure a robust design. Two types of imperfections are considered: variations of (1) the cross-sections and (2) the locations of structural elements. Both are modeled using random field theory.
The aim of this article is to introduce a new topology optimisation formulation for optimal robus... more The aim of this article is to introduce a new topology optimisation formulation for optimal robust design of Micro Electro Mechanical Systems. Mesh independence in topology optimisation is most often ensured by using filtering techniques, which result in transition grey regions difficult to interpret in practical realisations. This problem has been alleviated recently by projection techniques, but these destroy the mesh independence introduced by the filters and result in single node connected hinges. Such features in the design are undesirable as they are not robust with respect to geometric manufacturing errors (such as under/over etching). They can be avoided by optimising for several design realisations which take into account the possible geometry errors. The design variations are modelled with the help of random variables. The proposed stochastic formulation for the design variations results in nearly black and white mechanism designs, robust with respect to uncertainties in the production process, i.e. without any hinges or small details which can create manufacturing difficulties.
This article presents a topology optimization approach for micro- and nano-devices fabricated by ... more This article presents a topology optimization approach for micro- and nano-devices fabricated by optical projection lithography. Incorporating the photolithography process and the manufacturing uncertainties into the topology optimization process results in a binary mask that can be sent directly to manufacturing without additional optical proximity correction (OPC). The performance of the optimized device is robust toward the considered process variations. With the proposed unified approach, the design for photolithography is achieved by considering the optimal device performance and manufacturability at the same time. Only one optimization problem is solved instead of two as in the conventional separate procedures by (1) blueprint design and (2) OPC. A micro-gripper design example is presented to demonstrate the potential of this approach.
Structural and Multidisciplinary Optimization, 2014
This paper presents a flexible framework for parallel and easy-to-implement topology optimization... more This paper presents a flexible framework for parallel and easy-to-implement topology optimization using the Portable and Extendable Toolkit for Scientific Computing (PETSc). The presented framework is based on a standardized, and freely available library and in the published form it solves the minimum compliance problem on structured grids, using standard FEM and filtering techniques. For completeness a parallel implementation of the Method of Moving Asymptotes is included as well. The capabilities are exemplified by minimum compliance and homogenization problems. In both cases the unprecedented fine discretization reveals new design features, providing novel insight. The code can be downloaded from www.topopt.dtu.dk/PETSc.
Fourth International Workshop on Theoretical and Computational Nanophotonics (Tacona-Photonics 2011), 2011
The aim of this paper is to demonstrate 1D switch designs obtained by topology optimization which... more The aim of this paper is to demonstrate 1D switch designs obtained by topology optimization which show better performance than the designs considered in the literature. Such devices are non-linear and their performance depends on the efficiency of light-matter interaction. Simple optical switches can be designed using physical considerations and intuition. Alternatively the proposed topology optimization scheme provides a systematic methodology for obtaining and optimizing the layout of the devices. It is shown that the algorithm can efficiently handle more than two materials and that the obtained switches possess excellent performance.
Structural and Multidisciplinary Optimization, 2014
ABSTRACT This article presents a computational approach that facilitates the efficient solution o... more ABSTRACT This article presents a computational approach that facilitates the efficient solution of 3-D structural topology optimization problems on a standard PC. Computing time associated with solving the nested analysis problem is reduced significantly in comparison to other existing approaches. The cost reduction is obtained by exploiting specific characteristics of a multigrid preconditioned conjugate gradients (MGCG) solver. In particular, the number of MGCG iterations is reduced by relating it to the geometric parameters of the problem. At the same time, accurate outcome of the optimization process is ensured by linking the required accuracy of the design sensitivities to the progress of optimization. The applicability of the proposed procedure is demonstrated on several 2-D and 3-D examples involving up to hundreds of thousands of degrees of freedom. Implemented in MATLAB, the MGCG-based program solves 3-D topology optimization problems in a matter of minutes. This paves the way for efficient implementations in computational environments that do not enjoy the benefits of high performance computing, such as applications on mobile devices and plug-ins for modeling software.
Structural and Multidisciplinary Optimization, 2013
ABSTRACT The aim of this paper is to incorporate a model for micro/nano lithography production pr... more ABSTRACT The aim of this paper is to incorporate a model for micro/nano lithography production processes in topology optimization. The production process turns out to provide a physical analogy for projection filters in topology optimization. Blueprints supplied by the designers cannot be directly used as inputs to lithographic processes due to the proximity effect which causes rounding of sharp corners and geometric interaction of closely spaced design elements. Therefore, topology optimization is applied as a tool for proximity effect correction. Furthermore, it is demonstrated that the robust projection filter can be used to account for uncertainties due to lithographic production processes which results in manufacturable blueprint designs and eliminates the need for subsequent corrections.
Structural and Multidisciplinary Optimization, 2011
ABSTRACT Time domain topology optimization is applied to design pulse shaping filters. The object... more ABSTRACT Time domain topology optimization is applied to design pulse shaping filters. The objective function depends on the pulse envelope, which is extracted by utilizing the Hilbert transform. The gradients with respect to the topology optimization variables are derived, and the optimization methodology is demonstrated for pulse delaying and pulse splitting. The formulation is applicable for non-linear structures and signals consisting of broad range of frequencies. KeywordsTopology optimization–Hilbert transform–Envelope extraction
Structural and Multidisciplinary Optimization, 2011
This paper presents an efficient 88 line MATLAB code for topology optimization. It has been devel... more This paper presents an efficient 88 line MATLAB code for topology optimization. It has been developed using the 99 line code presented by Sigmund as a starting point. The original code has been extended by a density filter, and a considerable improvement in efficiency has been achieved, mainly by preallocating arrays and vectorizing loops. A speed improvement with a factor of 100 is achieved for a benchmark example with 7500 elements. Moreover, the length of the code has been reduced to a mere 88 lines. These improvements have been accomplished without sacrificing the readability of the code. The 88 line code can therefore be considered as a valuable successor to the 99 line code, providing a practical instrument that may help to ease the learning curve for those entering the field of topology optimization. The complete 88 line code is included as an appendix and can be downloaded from the web site www.topopt.dtu.dk. This is uncorrected version of the paper: Efficient topology optimization in MATLAB using 88 lines of code,
Structural and Multidisciplinary Optimization, 2011
Mesh convergence and manufacturability of topology optimized designs have previously mainly been ... more Mesh convergence and manufacturability of topology optimized designs have previously mainly been assured using density or sensitivity based filtering techniques. The drawback of these techniques has been gray transition regions between solid and void parts, but this problem has recently been alleviated using various projection methods. In this paper we show that simple projection methods do not ensure local mesh-convergence
Structural and Multidisciplinary Optimization, 2013
Topology optimization is a method to find the best distribution of material in a given design dom... more Topology optimization is a method to find the best distribution of material in a given design domain. The use of topology optimization for structural design often leads to slender structures which are sensitive to geometric imperfections such as the misplacement or misalignment of material. A robust approach to topology optimization is therefore presented which takes into account these geometric imperfections. An Eulerian approach is followed as the imperfections are modeled on the same finite element grid used in the deterministic topology optimization problem. Translation of material is obtained by adding a small perturbation to the center of the density filter kernel. The spatial variation of the geometric imperfections is modeled by means of a vector valued Gaussian random field. The random field is conditioned in order to incorporate supports in the design where no misplacement of material occurs. In the robust optimization problem, the objective function is defined as a weighted sum of the mean value and the standard deviation of the performance of the structure under uncertainty. A sampling method is used to estimate these statistics and the sensitivities thereof in the optimization algorithm. The solutions obtained by the robust approach are verified by means of an extensive Monte Carlo simulation.
Structural and Multidisciplinary Optimization, 2012
ABSTRACT The aim of this paper is to introduce the stochastic collocation methods in topology opt... more ABSTRACT The aim of this paper is to introduce the stochastic collocation methods in topology optimization for mechanical systems with material and geometric uncertainties. The random variations are modeled by a memory-less transformation of spatially varying Gaussian random fields which ensures their physical admissibility. The stochastic collocation method combined with the proposed material and geometry uncertainty models provides robust designs by utilizing already developed deterministic solvers. The computational cost is discussed in details and solutions to decrease it, like sparse grids and discretization refinement are proposed and demonstrated as well. The method is utilized in the design of compliant mechanisms.
The object of study is a stationary Gaussian white noise excited multi-degree-of-freedom (MDOF) l... more The object of study is a stationary Gaussian white noise excited multi-degree-of-freedom (MDOF) linear elastic, ideal plastic, linearly damped, statically determinate oscillator with several potential elements of ideal plastic yielding. Specifically the study is exemplified for a plane multistory shear frame with rigid traverses where all the connecting columns except the columns in one or more of the bottom floors have finite symmetrical yield limits. The white noise excitation acts on the mass of the first floor making the movement of the elastic bottom floors simulate a ground motion that interacts with the structure above the bottom floors. The method of study is so-called Slepian model simulation and is in principle the same for other statically determinate MDOF elasto-plastic oscillators of the considered type. The method is fast as compared to direct simulation and provides results that give good insight in the behavior of the plastic displacement response process. q
The object of study is a stationary Gaussian white noise excited plane multistory shear frame wit... more The object of study is a stationary Gaussian white noise excited plane multistory shear frame with a large number of rigid traverses. All the traverse-connecting columns have finite symmetrical yield limits except the columns in one or more of the bottom floors. The columns behave linearly elastic within the yield limits and ideally plastic outside these without accumulating eigenstresses. Within the elastic domain the frame is modeled as a linearly damped oscillator. The white noise excitation acts on the mass of the first floor making the movement of the elastic bottom floors simulate a ground motion that interacts with the structure above the bottom floors. As in a recent work by the authors the paper is about application of so-called Slepian model simulation, but in this paper supplemented by a simplification principle that allows a manageable calculation for the considered type of elasto-plastic oscillator also when it has a large number of elasto-plastic columns.
ABSTRACT We present a method to design manufacturable extremal elastic materials. Extremal materi... more ABSTRACT We present a method to design manufacturable extremal elastic materials. Extremal materials can possess interesting properties such as a negative Poisson’s ratio. The effective properties of the obtained microstructures are shown to be close to the theoretical limit given by mathematical bounds, and the deviations are due to the imposed manufacturing constraints. The designs are generated using topology optimization. Due to high resolution and the imposed robustness requirement they are manufacturable without any need for post-processing. This has been validated by the manufacturing of an isotropic material with a Poisson’s ratio of ν=-0.5ν=-0.5 and a bulk modulus of 0.2% times the solid base material’s bulk modulus.
International Journal for Numerical Methods in Engineering, 2012
ABSTRACT The aim of this paper was to present a topology optimization methodology for obtaining r... more ABSTRACT The aim of this paper was to present a topology optimization methodology for obtaining robust designs insensitive to small uncertainties in the geometry. The variations are modeled using a stochastic field. The model can represent spatially varying geometry imperfections in devices produced by etching techniques. Because of under-etching or over-etching parts of the structure may become thinner or thicker than a reference design supplied to the manufacturer. The uncertainties are assumed to be small and their influence on the system response is evaluated using perturbation techniques. Under the above assumptions, the proposed algorithm provides a computationally cheap alternative to previously introduced stochastic optimization methods based on Monte Carlo sampling. The method is demonstrated on the design of a minimum compliance cantilever beam and a compliant mechanism.
Computer Methods in Applied Mechanics and Engineering, 2015
This paper applies topology optimisation to the design of structures with periodic microstructura... more This paper applies topology optimisation to the design of structures with periodic microstructural details without length scale separation, i.e. considering the complete macroscopic structure and its response, while resolving all microstructural details, as compared to the often used homogenisation approach. The approach takes boundary conditions into account and ensures connected and macroscopically optimised microstructures regardless of the difference in micro-and macroscopic length scales. This results in microstructures tailored for specific applications rather than specific properties.
Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures, 2014
A probabilistic approach is presented to account for geometric imperfections in topology optimiza... more A probabilistic approach is presented to account for geometric imperfections in topology optimization in order to ensure a robust design. Two types of imperfections are considered: variations of (1) the cross-sections and (2) the locations of structural elements. Both are modeled using random field theory.
The aim of this article is to introduce a new topology optimisation formulation for optimal robus... more The aim of this article is to introduce a new topology optimisation formulation for optimal robust design of Micro Electro Mechanical Systems. Mesh independence in topology optimisation is most often ensured by using filtering techniques, which result in transition grey regions difficult to interpret in practical realisations. This problem has been alleviated recently by projection techniques, but these destroy the mesh independence introduced by the filters and result in single node connected hinges. Such features in the design are undesirable as they are not robust with respect to geometric manufacturing errors (such as under/over etching). They can be avoided by optimising for several design realisations which take into account the possible geometry errors. The design variations are modelled with the help of random variables. The proposed stochastic formulation for the design variations results in nearly black and white mechanism designs, robust with respect to uncertainties in the production process, i.e. without any hinges or small details which can create manufacturing difficulties.
This article presents a topology optimization approach for micro- and nano-devices fabricated by ... more This article presents a topology optimization approach for micro- and nano-devices fabricated by optical projection lithography. Incorporating the photolithography process and the manufacturing uncertainties into the topology optimization process results in a binary mask that can be sent directly to manufacturing without additional optical proximity correction (OPC). The performance of the optimized device is robust toward the considered process variations. With the proposed unified approach, the design for photolithography is achieved by considering the optimal device performance and manufacturability at the same time. Only one optimization problem is solved instead of two as in the conventional separate procedures by (1) blueprint design and (2) OPC. A micro-gripper design example is presented to demonstrate the potential of this approach.
Structural and Multidisciplinary Optimization, 2014
This paper presents a flexible framework for parallel and easy-to-implement topology optimization... more This paper presents a flexible framework for parallel and easy-to-implement topology optimization using the Portable and Extendable Toolkit for Scientific Computing (PETSc). The presented framework is based on a standardized, and freely available library and in the published form it solves the minimum compliance problem on structured grids, using standard FEM and filtering techniques. For completeness a parallel implementation of the Method of Moving Asymptotes is included as well. The capabilities are exemplified by minimum compliance and homogenization problems. In both cases the unprecedented fine discretization reveals new design features, providing novel insight. The code can be downloaded from www.topopt.dtu.dk/PETSc.
Fourth International Workshop on Theoretical and Computational Nanophotonics (Tacona-Photonics 2011), 2011
The aim of this paper is to demonstrate 1D switch designs obtained by topology optimization which... more The aim of this paper is to demonstrate 1D switch designs obtained by topology optimization which show better performance than the designs considered in the literature. Such devices are non-linear and their performance depends on the efficiency of light-matter interaction. Simple optical switches can be designed using physical considerations and intuition. Alternatively the proposed topology optimization scheme provides a systematic methodology for obtaining and optimizing the layout of the devices. It is shown that the algorithm can efficiently handle more than two materials and that the obtained switches possess excellent performance.
Structural and Multidisciplinary Optimization, 2014
ABSTRACT This article presents a computational approach that facilitates the efficient solution o... more ABSTRACT This article presents a computational approach that facilitates the efficient solution of 3-D structural topology optimization problems on a standard PC. Computing time associated with solving the nested analysis problem is reduced significantly in comparison to other existing approaches. The cost reduction is obtained by exploiting specific characteristics of a multigrid preconditioned conjugate gradients (MGCG) solver. In particular, the number of MGCG iterations is reduced by relating it to the geometric parameters of the problem. At the same time, accurate outcome of the optimization process is ensured by linking the required accuracy of the design sensitivities to the progress of optimization. The applicability of the proposed procedure is demonstrated on several 2-D and 3-D examples involving up to hundreds of thousands of degrees of freedom. Implemented in MATLAB, the MGCG-based program solves 3-D topology optimization problems in a matter of minutes. This paves the way for efficient implementations in computational environments that do not enjoy the benefits of high performance computing, such as applications on mobile devices and plug-ins for modeling software.
Structural and Multidisciplinary Optimization, 2013
ABSTRACT The aim of this paper is to incorporate a model for micro/nano lithography production pr... more ABSTRACT The aim of this paper is to incorporate a model for micro/nano lithography production processes in topology optimization. The production process turns out to provide a physical analogy for projection filters in topology optimization. Blueprints supplied by the designers cannot be directly used as inputs to lithographic processes due to the proximity effect which causes rounding of sharp corners and geometric interaction of closely spaced design elements. Therefore, topology optimization is applied as a tool for proximity effect correction. Furthermore, it is demonstrated that the robust projection filter can be used to account for uncertainties due to lithographic production processes which results in manufacturable blueprint designs and eliminates the need for subsequent corrections.
Structural and Multidisciplinary Optimization, 2011
ABSTRACT Time domain topology optimization is applied to design pulse shaping filters. The object... more ABSTRACT Time domain topology optimization is applied to design pulse shaping filters. The objective function depends on the pulse envelope, which is extracted by utilizing the Hilbert transform. The gradients with respect to the topology optimization variables are derived, and the optimization methodology is demonstrated for pulse delaying and pulse splitting. The formulation is applicable for non-linear structures and signals consisting of broad range of frequencies. KeywordsTopology optimization–Hilbert transform–Envelope extraction
Structural and Multidisciplinary Optimization, 2011
This paper presents an efficient 88 line MATLAB code for topology optimization. It has been devel... more This paper presents an efficient 88 line MATLAB code for topology optimization. It has been developed using the 99 line code presented by Sigmund as a starting point. The original code has been extended by a density filter, and a considerable improvement in efficiency has been achieved, mainly by preallocating arrays and vectorizing loops. A speed improvement with a factor of 100 is achieved for a benchmark example with 7500 elements. Moreover, the length of the code has been reduced to a mere 88 lines. These improvements have been accomplished without sacrificing the readability of the code. The 88 line code can therefore be considered as a valuable successor to the 99 line code, providing a practical instrument that may help to ease the learning curve for those entering the field of topology optimization. The complete 88 line code is included as an appendix and can be downloaded from the web site www.topopt.dtu.dk. This is uncorrected version of the paper: Efficient topology optimization in MATLAB using 88 lines of code,
Structural and Multidisciplinary Optimization, 2011
Mesh convergence and manufacturability of topology optimized designs have previously mainly been ... more Mesh convergence and manufacturability of topology optimized designs have previously mainly been assured using density or sensitivity based filtering techniques. The drawback of these techniques has been gray transition regions between solid and void parts, but this problem has recently been alleviated using various projection methods. In this paper we show that simple projection methods do not ensure local mesh-convergence
Structural and Multidisciplinary Optimization, 2013
Topology optimization is a method to find the best distribution of material in a given design dom... more Topology optimization is a method to find the best distribution of material in a given design domain. The use of topology optimization for structural design often leads to slender structures which are sensitive to geometric imperfections such as the misplacement or misalignment of material. A robust approach to topology optimization is therefore presented which takes into account these geometric imperfections. An Eulerian approach is followed as the imperfections are modeled on the same finite element grid used in the deterministic topology optimization problem. Translation of material is obtained by adding a small perturbation to the center of the density filter kernel. The spatial variation of the geometric imperfections is modeled by means of a vector valued Gaussian random field. The random field is conditioned in order to incorporate supports in the design where no misplacement of material occurs. In the robust optimization problem, the objective function is defined as a weighted sum of the mean value and the standard deviation of the performance of the structure under uncertainty. A sampling method is used to estimate these statistics and the sensitivities thereof in the optimization algorithm. The solutions obtained by the robust approach are verified by means of an extensive Monte Carlo simulation.
Structural and Multidisciplinary Optimization, 2012
ABSTRACT The aim of this paper is to introduce the stochastic collocation methods in topology opt... more ABSTRACT The aim of this paper is to introduce the stochastic collocation methods in topology optimization for mechanical systems with material and geometric uncertainties. The random variations are modeled by a memory-less transformation of spatially varying Gaussian random fields which ensures their physical admissibility. The stochastic collocation method combined with the proposed material and geometry uncertainty models provides robust designs by utilizing already developed deterministic solvers. The computational cost is discussed in details and solutions to decrease it, like sparse grids and discretization refinement are proposed and demonstrated as well. The method is utilized in the design of compliant mechanisms.
The object of study is a stationary Gaussian white noise excited multi-degree-of-freedom (MDOF) l... more The object of study is a stationary Gaussian white noise excited multi-degree-of-freedom (MDOF) linear elastic, ideal plastic, linearly damped, statically determinate oscillator with several potential elements of ideal plastic yielding. Specifically the study is exemplified for a plane multistory shear frame with rigid traverses where all the connecting columns except the columns in one or more of the bottom floors have finite symmetrical yield limits. The white noise excitation acts on the mass of the first floor making the movement of the elastic bottom floors simulate a ground motion that interacts with the structure above the bottom floors. The method of study is so-called Slepian model simulation and is in principle the same for other statically determinate MDOF elasto-plastic oscillators of the considered type. The method is fast as compared to direct simulation and provides results that give good insight in the behavior of the plastic displacement response process. q
The object of study is a stationary Gaussian white noise excited plane multistory shear frame wit... more The object of study is a stationary Gaussian white noise excited plane multistory shear frame with a large number of rigid traverses. All the traverse-connecting columns have finite symmetrical yield limits except the columns in one or more of the bottom floors. The columns behave linearly elastic within the yield limits and ideally plastic outside these without accumulating eigenstresses. Within the elastic domain the frame is modeled as a linearly damped oscillator. The white noise excitation acts on the mass of the first floor making the movement of the elastic bottom floors simulate a ground motion that interacts with the structure above the bottom floors. As in a recent work by the authors the paper is about application of so-called Slepian model simulation, but in this paper supplemented by a simplification principle that allows a manageable calculation for the considered type of elasto-plastic oscillator also when it has a large number of elasto-plastic columns.
ABSTRACT We present a method to design manufacturable extremal elastic materials. Extremal materi... more ABSTRACT We present a method to design manufacturable extremal elastic materials. Extremal materials can possess interesting properties such as a negative Poisson’s ratio. The effective properties of the obtained microstructures are shown to be close to the theoretical limit given by mathematical bounds, and the deviations are due to the imposed manufacturing constraints. The designs are generated using topology optimization. Due to high resolution and the imposed robustness requirement they are manufacturable without any need for post-processing. This has been validated by the manufacturing of an isotropic material with a Poisson’s ratio of ν=-0.5ν=-0.5 and a bulk modulus of 0.2% times the solid base material’s bulk modulus.
International Journal for Numerical Methods in Engineering, 2012
ABSTRACT The aim of this paper was to present a topology optimization methodology for obtaining r... more ABSTRACT The aim of this paper was to present a topology optimization methodology for obtaining robust designs insensitive to small uncertainties in the geometry. The variations are modeled using a stochastic field. The model can represent spatially varying geometry imperfections in devices produced by etching techniques. Because of under-etching or over-etching parts of the structure may become thinner or thicker than a reference design supplied to the manufacturer. The uncertainties are assumed to be small and their influence on the system response is evaluated using perturbation techniques. Under the above assumptions, the proposed algorithm provides a computationally cheap alternative to previously introduced stochastic optimization methods based on Monte Carlo sampling. The method is demonstrated on the design of a minimum compliance cantilever beam and a compliant mechanism.
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