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.@en

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.@en

Typically, the transient thermal error accounts for up to 70 percent of the total error in machine tools and precision devices. To decrease the effect of thermal distortions, design optimization is used as tool to modify, e.g., the material thickness, structural shape and cooling. This paper presents a heuristic design optimization method relying on thermal modes. The method reduces the system using the most relevant modes, which are identified using three designer-based response functions. Although the method may not converge to the global optimum and may need some tuning of inputs, it results in manufacturable designs. In addition, the designer-based response functions give insight in the design. Thus, this heuristic optimization method is a good way to reduce the thermal error in the design of a thermo-mechanical system. Keywords: thermal error, thermo-mechanical, heuristic design, design optimization, thermal modal analysis@en

Design sensitivities are used in various engineering methods to arrive at optimal or advanced designs. The design sensitivity of a system, also known as the derivative to a design variable, represents the change of the system response due to an adjustment of a design variable. Scientists and engineers often use modal design sensitivities to predict the change of a transient response. These sensitivities are the derivatives of the eigenvalues and eigenvectors of the system. Many researchers have successfully computed modal sensitivities using several methods. Nelson's method [1] is widely used even though computationally expensive. Fox's method [2] uses a modal superposition to represent the eigenvector sensitivities by the system eigenvectors, but is inaccurate when using only a few eigenvectors. Hence, several iterative methods were proposed to increase the accuracy of Fox's method (e.g., [3] or [4]). The primary focus of the previous research was on the computational algorithms rather than the interpretation of the eigenvector sensitivities. This interpretation depends highly on the chosen normalization. An approach for incorporating these normalizations was presented by Smith [5]. This paper shows the consequence and potential use for interpretation of the chosen normalization. This is illustrated on an example using a generic two-degrees-of-freedom system, as shown in FIGURE 2.@en

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.@en

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.@en

Reduced thermomechanical models are needed for the next generation precision devices. One of the methods is thermal modal analysis, which decomposes the behaviour into thermal modes and time constants. This study aims to measure these modal parameters. We isolated the behaviour of a single thermal mode exciting the system in the test set-up by this mode. The measured responses as well as the modal contributions are in good agreement with the simulations. Thus, thermal modal analysis can provide a basis for design optimiztion and compensation strategies.@en