This paper addresses the topology optimization of thermocouples for cooling applications, considering stress constraints to enhance reliability under service loads. We provide a first approach to derive sensitivities using SIMP (solid isotropic material with penalization) for thermo-electro-mechanical systems with temperature-dependent material properties. The proposed formulation decouples the thermoelectrical system from the mechanical degrees of freedom reducing computational memory usage from a fully coupled approach. The study focuses on the formulation of thermocouples for cooling applications using the Peltier effect, which considers electrical power limits, electrical working points, and material stress thresholds. Furthermore, while the thermoelectrical problem does not show the need for filtering techniques, including the mechanical degrees of freedom, we show that we recover undesirable porous optimized designs. We provide 2D thermocouple example optimizations with geometries and boundary conditions based on a practical case for the implementation of thermoelectric coolers in the Minimum Ionizing Particle Timing Detector (MTD) at CERN. The optimizations are performed with increased complexity, including the unfiltered thermoelectrical and thermo-electro-mechanical problems and a Helmholtz-filtered examples. The optimizations are compared with constant and nonlinear material properties with temperature and with respect to the consideration of air-conductance losses within the devices. Although more efficient topologies can be achieved without the need for volume constraints, we include an example with a constraint of 60% volume to understand its effect on the design and provide a methodology to reduce semiconductor-associated costs at lower efficiency costs. Finally, we explore the same formulation in 3D. The results provide guidelines for manufacturing compliant thermocouples, increasing their reliability without decreasing efficiency.

Residual stresses and distortions are major barriers to the broader adoption of wire arc additive manufacturing. These issues are coupled and arise due to large thermal gradients and phase transformations during the directed energy deposition process. Mitigating distortions may lead to substantial residual stresses, causing cracks in the fabricated components. In this paper, we propose a novel method to reduce both residual stresses and distortions by optimizing the fabrication sequence. This approach explores the use of non-planar layers, leveraging the increased manufacturing flexibility provided by robotic arms. Additionally, our method allows for the concurrent optimization of the structural layout and corresponding fabrication sequence. We employ the inherent strain method as a simplified process simulation model to predict residual stresses and distortions. Local residual stresses are aggregated using a p-norm function, which is integrated into distortion minimization as a constraint. Through numerical examples, we demonstrate that the optimized non-planar fabrication strategies can effectively reduce both residual stresses and distortions.

We propose a topology optimization (TO) formulation and related optimization scheme for designing compliant mechanisms following a user-defined trajectory. To ensure the broad applicability and achieve precisely control of the outputs, geometric nonlinearity with incremental solutions are considered. A challenge in the design optimization of these structures is the development of formulations with satisfactory balance between (i) precise trajectory control and (ii) proper connectivity between the input/output ports and the support. Previously proposed density-based topology optimization formulations typically lack the promotion of the desired load-transferring connections, or usually complicate the design using mixed shape, size, and topology variables to enforce a minimum connectivity. To simplify design progress using exclusive topology variables, i.e., purely density-based TO methods, we propose a relatively straightforward formulation involving commonly used response functions, such as compliance and volume as constraints. For the constraints, the paper provides a scheme for defining corresponding upper limits. Numerical examples of challenging shell and plate design optimization problems demonstrate the effectiveness of the proposed formulation and scheme in the generation of load-transferring connections while limiting the impact on the performance of the path generation functionality.

Inorganic scintillators often use exotic, expensive materials to increase their light yield. Although material chemistry is a valid way to increase the light collection, these methods are expensive and limited to the material properties. As such, alternative methods such as the use of specific reflective coatings and crystal optical shapes are critical for the scintillator crystal design procedure. In this paper, we explore the modeling of a scintillator and silicon-photomultiplier (SiPM) assembly detector using GEANT4. GEANT4, an open-source software for particle–matter interaction based on ray-tracing, allows the modeling of a scintillator-based detector while offering methods to simplify and study the computational requirements for a precise calculation of the light collection. These studies incorporate two different geometries compatible with the barrel timing layer (BTL) particle detector that is being built for the compact muon solenoid (CME) experiment at CERN. Furthermore, the geometry of our model is parameterized using splines for smoother results and meshed using GMSH to perform genetic numerical optimization of the crystal shape through genetic algorithms, in particular non-dominated sorting genetic algorithm II (NGSAII). Using NSGA-II, we provide a series of optimized scintillator geometries and study the trade-offs of multiple possible objective functions including the light output, light collection, light collection per energy deposited, and track path length. The converged Pareto results according to the hypervolume indicator are compared to the original simplified design, and a recommendation towards the use of the light collection per energy deposition and track path length is given based on the results. The results provide increases in this objective of up to 18% for a constant volume for a geometry compatible with the current design of the BTL detector.

Wire and Arc Additive Manufacturing (WAAM) has great potential for efficiently producing large metallic components. However, like other additive manufacturing techniques, materials processed by WAAM exhibit anisotropic properties. Assuming isotropic material properties in design optimization thus leads to less efficient material utilization. Instead of viewing WAAM-induced material anisotropy as a limitation, we consider it an opportunity to improve structural performance. This requires the integration of process planning into structural design. In this paper, we propose a novel method to utilize material anisotropy to enhance the performance of structures both during fabrication and in their use. Our approach is based on space–time topology optimization, which simultaneously optimizes the structural layout and the fabrication sequence. To model material anisotropy in space–time topology optimization, we derive the material deposition direction from the gradient of the pseudo-time field, which encodes the fabrication sequence. Numerical results demonstrate that leveraging material anisotropy effectively improves the performance of intermediate structures during fabrication as well as the overall structure.

In topology optimization of transient problems, memory requirements and computational costs often become prohibitively large due to the backward-in-time adjoint equations. Common approaches such as the Checkpointing (CP) and Local-in-Time (LT) algorithms reduce memory requirements by dividing the temporal domain into intervals and by computing sensitivities on one interval at a time. The CP algorithm reduces memory by recomputing state solutions instead of storing them. This leads to a significant increase in computational cost. The LT algorithm introduces approximations in the adjoint solution to reduce memory requirements and leads to a minimal increase in computational effort. However, we show that convergence can be hampered using the LT algorithm due to errors in approximate adjoints. To reduce memory and/or computational time, we present two novel algorithms. The hybrid Checkpointing/Local-in-Time (CP/LT) algorithm improves the convergence behavior of the LT algorithm at the cost of an increased computational time but remains more efficient than the CP algorithm. The Parallel-Local-in-Time (PLT) algorithm reduces the computational time through a temporal parallelization in which state and adjoint equations are solved simultaneously on multiple intervals. State and adjoint fields converge concurrently with the design. The effectiveness of each approach is illustrated with two-dimensional density-based topology optimization problems involving transient thermal or flow physics. Compared to the other discussed algorithms, we found a significant decrease in computational time for the PLT algorithm. Moreover, we show that under certain conditions, due to the use of approximations in the LT and PLT algorithms, they exhibit a bias toward designs with short characteristic times. Finally, based on the required memory reduction, computational cost, and convergence behavior of optimization problems, guidelines are provided for selecting the appropriate algorithms.

In additive manufacturing, the fabrication sequence has a large influence on the quality of manufactured components. While planning of the fabrication sequence is typically performed after the component has been designed, recent developments have demonstrated the possibility and benefits of simultaneous optimization of both the structural layout and the corresponding fabrication sequence. This is particularly relevant in multi-axis additive manufacturing, where rotational motion offers enhanced flexibility compared to planar fabrication. The simultaneous optimization approach, called space–time topology optimization, introduces a pseudo-time field to encode the manufacturing process order, alongside a pseudo-density field representing the structural layout. To comply with manufacturing principles, the pseudo-time field needs to be monotonic, i.e., free of local minima. However, explicitly formulated constraints proposed in prior work are not always effective, particularly for complex structural layouts that commonly result from topology optimization. In this paper, we introduce a novel method to regularize the pseudo-time field in space–time topology optimization. We conceptualize the monotonic additive manufacturing process as a virtual heat conduction process starting from the surface upon which a component is constructed layer by layer. The virtual temperature field, which shall not be confused with the actual temperature field during manufacturing, serves as an analogy for encoding the fabrication sequence. In this new formulation, we use local virtual heat conductivity coefficients as optimization variables to steer the temperature field and, consequently, the fabrication sequence. The virtual temperature field is inherently free of local minima due to the physics it resembles. We numerically validate the effectiveness of this regularization in space–time topology optimization under process-dependent loads, including gravity and thermomechanical loads.

Heat pumping through thermoelectric devices has many advantages over traditional cooling. However, their current efficiency is a limiting factor in their implementation. In this paper, we approach the non-convex topology optimization of thermoelectrical elements for cooling applications through the method of moving asymptotes (MMA) to improve their cooling capabilities per watt usage. The optimization problem is defined for a given power budget, aiming for the minimum temperature with a known heat pumping need. The introduction of power as a constraint justifies the introduction of the voltage gradient across the thermocouple as a design variable to maintain the thermoelectrical device in its optimum power-to-heat extraction ratio. To better understand the convergence of this non-convex problem, we present a two-variable analytical thermoelectric optimization model. This example provides information on how to select the penalty parameters used to scale the three material coefficients involved in the problem to obtain lower objective values and better convergence using MMA. The analytical model shows the non-convexity of the problem and provides the recommendation to use penalization coefficients of the form p_k=p_σ>p_α=1 for the thermal conductivity, electrical conductivity, and Seebeck coefficients. We tested these penalization coefficients through optimizations of a model based on the 1MC10-031 commercial thermoelectric-cooler (TEC) using the finite element method (FEM). These penalization coefficients provided local minima without the need for volume constraints. With this procedure, we found designs that provided temperatures close to 10 degrees lower using 60% less semiconductor material volume compared to the initial design.

Modeling of fluid flows in density-based topology optimization forms a longstanding challenge. Methods based on the Navier–Stokes equations with Darcy penalization (NSDP equations) are widely used in fluid topology optimization. These methods use porous materials with low permeability to represent the solid domain. Consequently, they suffer from flow leakage in certain areas. In this work, the governing equations for solid/fluid density-based topology optimization are reevaluated and reinterpreted. The governing equations are constructed using the volume averaged Navier–Stokes (VANS) equations, well known in the field of porous flow modeling. Subsequently, we simplify, interpret and discretize the VANS equations in the context of solid/fluid topology optimization, and analytically derive lower bounds on the Darcy penalization to sufficiently prevent flow leakage. Based on both the NSDP and VANS equations, two flow solvers are constructed using the Finite Volume method. Their precision and the lower bound on the Darcy penalization are investigated. Subsequently, the solvers are used to optimize flow channels for minimal pressure drop, and the resulting designs and convergence behavior are compared. The optimization procedure using the VANS equations is found to show less tendency to converge to inferior local optima for more precise flow solutions and is less sensitive to its parameter selection.

Overheating is a major issue especially in metal Additive Manufacturing (AM) processes, leading to poor surface quality, lack of dimensional precision, inferior performance and/or build failures. A 3D density-based topology optimization (TO) method is presented which addresses the issue of local overheating during metal AM. This is achieved by integrating a simplified AM thermal model and a thermal constraint within the optimization loop. The simplified model, recently presented in literature, offers significant computational gains while preserving the ability of overheating detection. The novel thermal constraint ensures that the overheating risk of optimized designs is reduced. This is fundamentally different from commonly used geometry-based TO methods which impose a geometric constraint on overhangs. Instead, the proposed approach takes the process physics into account. The proposed method is validated via an experimental comparative study. Optical tomography (OT) is used for in-situ monitoring of process conditions during fabrication and obtained data is used for evaluation of overheating tendencies. The novel TO method is compared with two other methods: standard TO and TO with geometric overhang control. The experimental data reveals that the novel physics-based TO design experienced less overheating during the build as compared to the two classical designs. A study further investigated the correlation between overheating observed by high OT values and the defect of porosity. It shows that overheated regions indeed show higher defect of porosity. This suggests that geometry-based guidelines, although enhance printability, may not be sufficient for eliminating overheating issues and related defects. Instead, the proposed physics-based method is able to deliver efficient designs with reduced risk of overheating.

Additive manufacturing of metal parts involves phase transformations and high temperature gradients which lead to uneven thermal expansion and contraction, and, consequently, distortion of the fabricated components. The distortion has a great influence on the structural performance and dimensional accuracy, e.g., for assembly. It is therefore of critical importance to model, predict and, ultimately, reduce distortion. In this paper, we present a computational framework for fabrication sequence optimization to minimize distortion in multi-axis additive manufacturing (e.g., robotic wire arc additive manufacturing), in which the fabrication sequence is not limited to planar layers only. We encode the fabrication sequence by a continuous pseudo-time field, and optimize it using gradient-based numerical optimization. To demonstrate this framework, we adopt a computationally tractable yet reasonably accurate model to mimic the material shrinkage in metal additive manufacturing and thus to predict the distortion of the fabricated components. Numerical studies show that optimized curved layers can reduce distortion by orders of magnitude as compared to their planar counterparts.

The published article contains a mistake in the notation of the strain–displacement matrix (Formula presented.) in Voigt notation concerning the isoparametric mapping and in the notation of the corresponding derivatives. As the mistakes relate only to the notation, and the implementation in the code was done correctly, they bear no consequences for the remainder of the manuscript and the results presented therein. The (Formula presented.) matrix defined inline after Eq. (11) on page 5 (repeated on pages 8 and 17) should be written as (Formula presented.) for an enriched element with n original shape functions and m enrichment functions. The differential operator (Formula presented.) in Eq. 12 should then have been defined in Voigt notation and in global coordinates as (Formula presented.) for elastostatics in 2-D and 3-D, respectively, and (Formula presented.) for heat conductivity in 2-D and 3-D, respectively. The derivatives in global coordinates are computed from the derivatives in local coordinates as (Formula presented.) for standard and enriched shape functions, respectively, where (Formula presented.) is the Jacobian of the intersected original element and (Formula presented.) is the Jacobian of the integration element. Furthermore, the derivative of (Formula presented.) with respect to the enriched node location (Formula presented.) in Equation 28 should be written as (Formula presented.) Note that the terms in (Formula presented.) corresponding to the standard shape functions are zero (this equation replaces Eq. 29 of the manuscript): (Formula presented.) as the Jacobian of the parent element is not influenced by the enriched node location, and the derivatives of the standard shape functions with respect to local coordinates of the parent element are constant. The components in (Formula presented.) corresponding to the enrichment functions are computed using (this equation replaces Eq. 30 in the manuscript) (Formula presented.) where the second term is zero because the derivatives of the enrichment function with respect to local coordinates are constant and independent of the enriched node location in global coordinates.

Phononic crystals can be designed to show band gaps—ranges of frequencies whose propagation is strongly attenuated in the material. In essence, their working principle is based on destructive interference of waves reflecting from the periodic arrangement of material interfaces (i.e., Bragg scattering). Consequently, capturing accurately the behavior at material interfaces requires appropriate numerical modeling and computational design techniques. However, the commonly used density-based representation in popular topology optimization methods results in a diffuse staircased boundary. The heavily refined finite element meshes required to compensate for this boundary description results in exceedingly large and expensive optimization problems. In this paper, we demonstrate the adverse effect of the density-based boundary description. Furthermore, we propose a level set-based topology optimization procedure with an enriched finite element method that shows improved performance when compared to the density-based approach.

The design of high-performance mechatronic systems is very challenging, as it requires delicate balancing of system dynamics, the controller, and their closed-loop interaction. Topology optimization provides an automated way to obtain systems with superior performance, although extension to simultaneous optimization of both topology and controller has been limited. To allow for topology optimization of mechatronic systems for closed-loop performance, stability, and disturbance rejection (i.e. modulus margin), we introduce local approximations of the Nyquist curve using circles. These circular approximations enable simple geometrical constraints on the shape of the Nyquist curve, which is used to characterize the closed-loop performance. Additionally, a computationally efficient robust formulation is proposed for topology optimization of dynamic systems. Based on approximation of eigenmodes for perturbed designs, their dynamics can be described with sufficient accuracy for optimization, while preventing the usual threefold increase of additional computational effort. The designs optimized using the integrated approach have significantly better performance (up to 350% in terms of bandwidth) than sequentially optimized systems, where eigenfrequencies are first maximized and then the controller is tuned. The proposed approach enables new directions of integrated (topology) optimization, with effective control over the Nyquist curve and efficient implementation of the robust formulation.

High computational costs are encountered in topology optimization problems of geometrically nonlinear structures since intensive use has to be made of incremental-iterative finite element simulations. To alleviate this computational intensity, reduced-order models (ROMs) are explored in this paper. The proposed method targets ROM bases consisting of a relatively small set of base vectors while accuracy is still guaranteed. For this, several fully automated update and maintenance techniques for the ROM basis are investigated and combined. In order to remain effective for flexible structures, path derivatives are added to the ROM basis. The corresponding sensitivity analysis (SA) strategies are presented and the accuracy and efficiency are examined. Various geometrically nonlinear examples involving both solid as well as shell elements are studied to test the proposed ROM techniques. Test cases demonstrates that the set of degrees of freedom appearing in the nonlinear equilibrium equation typically reduces to several tenth. Test cases show a reduction of up to 6 times fewer full system updates.

Computational process modelling of metal additive manufacturing has gained significant research attention in recent past. The cornerstone of many process models is the transient thermal response during the AM process. Since deposition-scale modelling of the thermal conditions in AM is computationally expensive, spatial and temporal simplifications, such as simulating deposition of an entire layer or multiple layers, and extending the laser exposure times, are commonly employed in the literature. Although beneficial in reducing computational costs, the influence of these simplifications on the accuracy of temperature history is reported on a case-by-case basis. In this paper, the simplifications from the existing literature are first classified in a normalised simplification space based on assumptions made in spatial and temporal domains. Subsequently, all types of simplifications are investigated with numerical examples and compared with a high-fidelity reference model. The required numerical discretisation for each simplification is established, leading to a fair comparison of computational times. The holistic approach to the suitability of different modelling simplifications for capturing thermal history provides guidelines for the suitability of simplifications while setting up a thermal AM model.

In recent years, the Q-factor of Si 3 N 4 nanomechanical resonators has significantly been increased by soft-clamping techniques using large and complex support structures. To date, however, obtaining similar performance with smaller supports has remained a challenge. Here, we make use of torsion beam supports to tune the Q-factor of Si 3 N 4 string resonators. By design optimization of the supports, we obtain a 50% Q-factor enhancement compared to the standard clamped-clamped string resonators. By performing experimental and numerical studies, we show that further improvement of the Q-factor is limited by a trade-off between maximizing stress and minimizing torsional support stiffness. Thus, our study also provides insight into dissipation limits of high-stress string resonators and outlines how advanced designs can be realized for reaching ultimate f 0 × Q product while maintaining a small footprint.

A novel constraint to prevent local overheating is presented for use in topology optimization (TO). The very basis for the constraint is the Additive Manufacturing (AM) process physics. AM enables fabrication of highly complex topologically optimized designs. However, local overheating is a major concern especially in metal AM processes leading to part failure, poor surface finish, lack of dimensional precision, and inferior mechanical properties. It should therefore be taken into account at the design optimization stage. However, including a detailed process simulation in the optimization would make the optimization intractable. Hence, a computationally inexpensive thermal process model, recently presented in the literature, is used to detect zones prone to local overheating in a given part geometry. The process model is integrated into density-based TO in combination with a robust formulation, and applied in various numerical test examples. It is found that existing AM-oriented TO methods which rely purely on overhang control do not ensure overheating avoidance. Instead, the proposed physics-based constraint is able to suppress geometric features causing local overheating and delivers optimized results in a computationally efficient manner.

In topology optimization, the state of structures is typically obtained by numerically evaluating a discretized PDE-based model. The degrees of freedom of such a model can be partitioned in free and prescribed sets to define the boundary conditions. A multi-partition problem involves multiple partitions of the same discretization, typically corresponding to different loading scenarios. As a result, solving multi-partition problems involves multiple factorization/preconditionings of the system matrix, requiring a high computational effort. In this paper, a novel method is proposed to efficiently calculate the responses and accompanying design sensitivities in such multi-partition problems using static condensation for use in gradient-based topology optimization. A main problem class that benefits from the proposed method is the topology optimization of small-displacement multi-input–multi-output compliant mechanisms. However, the method is applicable to any linear problem. We present its formulation and an algorithmic complexity analysis to estimate computational advantages for both direct and iterative solution methods to solve the system of equations, verified by numerical experiments. It is demonstrated that substantial gains are achievable for large-scale multi-partition problems. This is especially true for problems with both a small set of number of degrees of freedom that fully describes the performance of the structure and with large similarities between the different partitions. A major contribution to the gain is the lack of large adjoint analyses required to obtain the sensitivities of the performance measure.

High-tech equipment critically relies on flexures for precise manipulation and measurement. Through elastic deformation, flexures offer extreme position repeatability within a limited range of motion in their degrees of freedom, while constraining motion in the degrees of constraint. Topology optimization proves a prospective tool for the design of short-stroke flexures, providing maximum design freedom and allowing for application-specific requirements. State-of-the-art topology optimization formulations for flexure synthesis are subject to challenges like ease of use, versatility, implementation complexity, and computational cost, leaving a generally accepted formulation absent. This study proposes a novel topology optimization formulation for the synthesis of short-stroke flexures uniquely based on strain energy measures under prescribed displacement scenarios. The resulting self-adjoint optimization problem resembles great similarity to ‘classic’ compliance minimization and inherits similar implementation simplicity, computational efficiency, and convergence properties. Numerical examples demonstrate the versatility in flexure types and the extendability of additional design requirements. The provided source code encourages the formulation to be explored and applied in academia and industry.