EH
E.C. Hooijkamp
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Topology optimization for linear thermo-mechanical transient problems
Modal reduction and adjoint sensitivities
This paper focuses on topology optimization for linear transient thermo-mechanical problems. The latter are, for example, encountered for extreme precision tools and instrumentation. Due to the transient nature, a standard adjoint sensitivity analysis will result in a backward transient analysis for the adjoint variables, leading to both storage and computational inefficiencies. A method is proposed that rigorously eliminates the backward transient integration for the adjoint sensitivity analysis. At the basis is a model-order reduction technique, which relies on a reduced thermal modal basis combined with static correction. The modal amplitudes can be readily obtained semi-analytically using simple convolutions. This accurate but reduced-order model is the starting point for an adjoint sensitivity analysis. Via a tactic selection of adjoint variables, the backward transient analysis for the adjoints is completely eliminated, whereas computational efficiency and consistency are maintained. The effectiveness of the resulting adjoint sensitivities and their application in topology optimization are demonstrated on the basis of several test examples.
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This paper focuses on topology optimization for linear transient thermo-mechanical problems. The latter are, for example, encountered for extreme precision tools and instrumentation. Due to the transient nature, a standard adjoint sensitivity analysis will result in a backward transient analysis for the adjoint variables, leading to both storage and computational inefficiencies. A method is proposed that rigorously eliminates the backward transient integration for the adjoint sensitivity analysis. At the basis is a model-order reduction technique, which relies on a reduced thermal modal basis combined with static correction. The modal amplitudes can be readily obtained semi-analytically using simple convolutions. This accurate but reduced-order model is the starting point for an adjoint sensitivity analysis. Via a tactic selection of adjoint variables, the backward transient analysis for the adjoints is completely eliminated, whereas computational efficiency and consistency are maintained. The effectiveness of the resulting adjoint sensitivities and their application in topology optimization are demonstrated on the basis of several test examples.
Design of transient thermo-mechanical systems is a challenging task often encountered during the design of high precision machines and instrumentation. Topology optimization can provide valuable insight during the design process, however, for large scale problems the backward time integration required to obtain adjoint sensitivity information is undesired. Previous work has illustrated how the introduction of a reduced modal basis allows to eliminate the backward time integration to obtain the adjoint variables. In order to reduce computational effort further, additional reduction approaches are considered. The focus is specifically on design cases where the relevant heat loads can be expressed or approximated analytically by combinations of harmonic, polynomial or exponential functions of time. Using the method of undetermined coefficients, an exact particular solution is obtained using the full system. Then, the corresponding homogenous solution is expressed using a reduced modal basis, for which a relatively small set of modes is required to obtain an accurate approximation. For the cases where the time component of the heat loads are expressed by the considered analytical functions, the backward time integration is eliminated from the calculation of the design sensitivities, while the forward integration is handled by convolutions.
...
Design of transient thermo-mechanical systems is a challenging task often encountered during the design of high precision machines and instrumentation. Topology optimization can provide valuable insight during the design process, however, for large scale problems the backward time integration required to obtain adjoint sensitivity information is undesired. Previous work has illustrated how the introduction of a reduced modal basis allows to eliminate the backward time integration to obtain the adjoint variables. In order to reduce computational effort further, additional reduction approaches are considered. The focus is specifically on design cases where the relevant heat loads can be expressed or approximated analytically by combinations of harmonic, polynomial or exponential functions of time. Using the method of undetermined coefficients, an exact particular solution is obtained using the full system. Then, the corresponding homogenous solution is expressed using a reduced modal basis, for which a relatively small set of modes is required to obtain an accurate approximation. For the cases where the time component of the heat loads are expressed by the considered analytical functions, the backward time integration is eliminated from the calculation of the design sensitivities, while the forward integration is handled by convolutions.
Journal article
(2016)
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Rudolf Saathof, M.V. Wansink, Evert Hooijkamp, Jo Spronck, Robert Munnig Schmidt
A demonstrator adaptive optics-system with a thermally actuated active mirror (AM) is presented to pre-study feasibility of sub-nm wavefront control in extreme ultraviolet (EUV) lithography. The AM is thermally actuated by selective heating using a spatial controllable heat source. Four different methods have been implemented to control the deformation of the AM. First thermal feedforward using estimated state feedback (ESF), second thermal feedback using proportional integral (PI) control, third their combination and fourth deformation feedback using PI control. To support ESF, a thermo-elastic finite element model is employed that describes the thermal deformation of the AM. ESF shows satisfying performance with a time constant of 10 s and a residual error of 0.7 nm. Thermal feedback shows large fluctuations of 12 nm for spherical aberrations of due to feedback of noise from the thermal camera. By applying deformation feedback the RMS-error is reduced to a satisfying 0.25 nm. This study shows that deformation control of this AM can be realised using thermal feedforward and deformation feedback to meet the requirements for EUV lithography.
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A demonstrator adaptive optics-system with a thermally actuated active mirror (AM) is presented to pre-study feasibility of sub-nm wavefront control in extreme ultraviolet (EUV) lithography. The AM is thermally actuated by selective heating using a spatial controllable heat source. Four different methods have been implemented to control the deformation of the AM. First thermal feedforward using estimated state feedback (ESF), second thermal feedback using proportional integral (PI) control, third their combination and fourth deformation feedback using PI control. To support ESF, a thermo-elastic finite element model is employed that describes the thermal deformation of the AM. ESF shows satisfying performance with a time constant of 10 s and a residual error of 0.7 nm. Thermal feedback shows large fluctuations of 12 nm for spherical aberrations of due to feedback of noise from the thermal camera. By applying deformation feedback the RMS-error is reduced to a satisfying 0.25 nm. This study shows that deformation control of this AM can be realised using thermal feedforward and deformation feedback to meet the requirements for EUV lithography.
Designing transient thermal mechanical systems is a challenging task. Material can have many different functions: it can provide heat capacity, heat conduction, mechanical stiffness or even function as an actuator. Topology optimization can provide the engineer with valuable insight on such a problem. One of the most popular topology optimization approaches is the density method. This method is applied to a transient thermal mechanical problem. In order to ensure manufacturability, penalization is applied to suppress intermediate densities in the final design. However, for transient thermal mechanical optimization problems, conventional penalization does not work for most objective functions. A new penalization method, material penalization, is presented that does suppress intermediate densities in the transient thermal mechanical domain. Each element is given its own unique set of penalization parameters which are optimized to maximize the objective function for a minimization problem. By reusing sensitivity information from the density variables, the additional computational cost is limited.
...
Designing transient thermal mechanical systems is a challenging task. Material can have many different functions: it can provide heat capacity, heat conduction, mechanical stiffness or even function as an actuator. Topology optimization can provide the engineer with valuable insight on such a problem. One of the most popular topology optimization approaches is the density method. This method is applied to a transient thermal mechanical problem. In order to ensure manufacturability, penalization is applied to suppress intermediate densities in the final design. However, for transient thermal mechanical optimization problems, conventional penalization does not work for most objective functions. A new penalization method, material penalization, is presented that does suppress intermediate densities in the transient thermal mechanical domain. Each element is given its own unique set of penalization parameters which are optimized to maximize the objective function for a minimization problem. By reusing sensitivity information from the density variables, the additional computational cost is limited.