The rising power density of modern electronics requires effective thermal management to maintain performance and reliability. Phase change materials (PCMs) provide passive cooling via latent heat but are limited by low thermal conductivity, which hinders their efficiency. This study investigates the application of topology optimization for designing thermally efficient fin structures in a PCM-filled domain. The PCM numerical model is validated against experimental data, ensuring code fidelity. The optimization process is conducted within a 2D finite element framework, where fins are designed on a fixed-temperature wall while maintaining the same material volume as a baseline configuration with three rectangular fins. Three objective functions - convective heat flux, diffusive heat flux, and thermal compliance - are considered using a steady-state approach. Their influence on the melting process is systematically evaluated. Then, the best performing one is used in a multi-step approach (MSTE) that resembles the unsteady PCM melting but allows to obtain performing design through an iterative process. Results demonstrate that the choice of a convective heat flux as objective accelerates the melting process by 32.1% compared to the baseline. Then, the MSTE shows that with two iterations it is possible to generate a design that fasten the melting process by 48.9% compared to the baseline. Further analysis includes comparisons of melt fraction, Nusselt number, energy storage and mean thermal power across all cases to provide valuable insights into the advantage of the MSTE approach for optimizing thermal management systems.

Flat miniature heat pipes (FMHPs) are compact, passive thermal management devices that leverage phase change to sustain high heat fluxes. Their performance is strongly influenced by the geometry of the internal wick structure, which governs capillary-driven liquid transport. This work presents a reduced-order numerical model for predicting the steady-state thermal performance of FMHPs with axially grooved wicks. The model captures key thermohydraulic phenomena while enabling efficient gradient-based shape optimization by simplifying the governing equations and linearizing the phase-change mass flux. The groove width is parametrized along the axial direction and optimized to maximize the heat transport capacity. Validation against experimental and high-fidelity numerical results demonstrates the model’s accuracy and efficiency. Optimization results show that non-uniform groove profiles can significantly enhance performance, achieving up to 24% improvement in maximum heat input compared to uniform grooves, without increasing the device footprint. The study highlights the potential of shape optimization in advancing the design of high-performance FMHPs for electronics cooling and other heat-intensive applications.

The mechanical performance of short‐fiber‐reinforced polymers produced by Material EXtrusion Additive Manufacturing (MEX‐AM) is strongly influenced by the fiber microstructure formed during deposition. This work introduces a manufacturing‐aware topology optimization framework that exploits process‐induced microstructure control to enhance structural performance. A three-dimensional computational fluid dynamics model of the material deposition process is used to predict the spatial evolution of fiber orientation as a function of nozzle rotation speed, revealing a continuous transition from highly aligned to nearly isotropic. Based on these simulations, an effective material model is constructed in which the local elastic properties depend directly on the chosen nozzle rotation speed. This model is integrated into a density‐based topology optimization scheme that simultaneously determines the structural layout, the spatially varying fiber orientations, and the local level of anisotropy. The nozzle rotation speed acts as a design variable that allows the optimizer to tailor the anisotropy ratio throughout the structure. Numerical examples show that the framework naturally assigns highly aligned fibers to regions dominated by uniaxial loading, while favoring microstructures with low anisotropy in areas experiencing multi‐axial or shear‐dominated stress states. The results demonstrate the potential of coupling process‐level microstructure prediction with topology optimization to design fiber‐reinforced components for MEX-AM with spatially programmed material behavior. The proposed approach establishes a general pathway for integrating manufacturing physics with design optimization in additively manufactured structures.

The objective of this work is to investigate the optimal distribution of solid material within the cavity that minimizes heat transfer across its vertical boundaries. To this end, we explore the application of topology optimization (TO) - a methodology used to determine the optimal layout of material within a given design domain - to address the challenge of thermal insulation in rectangular cavities subjected to differential heating. By strategically placing material, the aim is to balance the diffusion/convection trade-off between low-velocity air gaps and diffusive solid paths, improving the thermal resistance of the heated cavity. The optimization problem is set based on a conjugate heat transfer formulation, i.e., the governing equations of heat conduction inside solid material, and natural convection inside air domains. Regularization techniques are applied to ensure the obtained solutions possess certain desirable characteristics, e.g., smoothness and discreteness, essential for instance in additive manufacturing for features such as 3D printability. A dimensionless problem is firstly solved to explore how the material distribution changes with the air recirculation magnitude, i.e., changing the Rayleigh number (Ra) value from 105 to 108. Further, the impact of mesh size, objective function selection, and initial design is investigated. Parameters affecting the TO routine are thus calibrated based on these simulations. Resulting designs are compared with baseline solutions consisting of vertical partitions, fixing the same volume of solid material. Material distribution of TO cases strongly differs depending on the Ra value flow strength. Therefore, convective motions are locally obstructed by smart solid features, leading to reduction in the Nusselt number up to 26 % at Ra = 108

The optimization results of level-set methods typically suffer from a strong dependence on the initial design. To mitigate this dependence, this work presents a density-based hole nucleation method for level-set topology optimization of flow problems. This is achieved by defining both the level-set and density values as functions of the design variables. To preserve the crispness of the geometry definition afforded by the level-set method, the fluid flow is modeled using the Heaviside enriched eXtended Finite Element Method (XFEM) for the laminar incompressible Navier–Stokes equation, which is augmented by a Brinkman model. The boundary conditions are enforced weakly using Nitsche’s method. A face-oriented ghost stabilization scheme is applied to stabilize the XFEM formulation. Additional terms ensuring stability are added to Nitsche’s method and the ghost stabilization to account for the Brinkman term in the Navier–Stokes equation. The necessity of adding these terms is highlighted by numerical studies. Two- and three-dimensional fluid manifolds are optimized to minimize fluid power dissipation while achieving a predefined mass flow distribution among the outlets. The optimization results show that the proposed method bypasses the need for an initial hole seeding and speeds up the convergence of the optimization process.

Labyrinth seals are mechanical devices that reduce the leakage between moving components of rotating machines. They are crucial for determining the performance of these machines and essential for reducing greenhouse gas emissions. Therefore, mathematical optimization algorithms for labyrinth seals are of high interest. This work concerns the topology optimization of labyrinth seals, which is a challenging task because the optimizer must distribute solids at different velocities to properly model the flow in the cavity of the labyrinth seal. Also, the model must not allow the optimizer to close the fluid channel or to leave free-floating islands of solid elements in the final design. Therefore, the interface identification method based on a series of erosion-dilate projection filters is used to model the fluid channel as the interface between the rotor and stator, allowing the representation of different velocities and avoiding the channel closure. The virtual temperature method is employed as a connectivity constraint to avoid free-floating islands of solid material. Still, thin structural members appear in the optimized designs, so geometric constraints are used to impose the minimum length scale on rotor and stator components. The final topology optimization results are labyrinth seals with contorted channels constricted to the minimum allowed gap size (as expected for low Reynolds flows), showing the success of the proposed formulation. The staggered and stepped labyrinth seal configurations are optimized to show the generality of the method.

This paper presents a method for simultaneous optimization of the outer shape and internal topology of aircraft wings, with the objective of minimizing drag subject to lift and compliance constraints for multiple load cases. The physics are evaluated by the means of a source-doublet panel method for the aerodynamic response and linear elastic finite elements for the structural response, which are one-way coupled. At each design iteration, a mapping procedure is applied to map the current wing shape and corresponding pressure loads to the unfitted finite element mesh covering the design domain. Wings of small fixed-wing airplanes both with and without a stiffening strut are optimized. The resulting wings show internal topologies with struts and wall-truss combinations, depending on the design freedom of the shape optimization. The lift distributions of the optimized wings show patterns like the ones obtained when performing optimization of wing shapes with constraints on the bending moment at the root.

This work presents studies on static aeroelastic shape optimization of aircraft wings subject to large deformations. The physics are captured using a coupled 3D panel method and a nonlinear co-rotating beam finite element model. The wing is defined by a series of airfoils that are parameterized based on the definition of NACA 4-digit airfoils. The method assumes a solid cross section of isotropic material which is representative of foam core wings. Analytic expressions are derived for most of the cross-sectional stiffness properties, while approximations are introduced for the location of the shear center and the torsional stiffness. Optimized designs achieved using linear and nonlinear deformation models are compared and features are discussed. The objective is to minimize drag subject to constraints on geometry, tip displacement, and root bending moment. Results highlight the importance of using nonlinear models to accurately capture changes in wingspan due to large deformations, as even small differences in the wingspan can have a large effect on the induced drag. Non-planar wings with raised and drooped wingtips are also optimized, where drooped wings are found to achieve larger lift-to-drag ratios due to the increase in effective wingspan in the deformed configuration.

This paper demonstrates that there is much more to gain from topology optimization of heat sinks than what is described by the so-called pseudo 3D models. The utilization of 3D effects, even for microchannel heat sinks is investigated and compared to state-of-the art industrial designs, for a microelectronic application. Furthermore, the use of design restrictions in the optimization framework demonstrates that the performances of microchannel heat sinks are highly dependent on the ability to provide complex refrigerant distribution and intricate flow paths through the heat sink. The topology optimized microchannel heat sinks are exported from a voxel mesh to bodyfitted mesh using Trelis Sculpt and imported into a commercial CFD software. A systematic comparison with the state-of-the art industrial design shows that the temperature elevation of the microelectronic chip can be reduced by up to 70%, using a 3D topology optimized microchannel heat sink. Restricting the design freedom, for example, by limiting the solid features to be unidirectional downgrades the performances of the optimized microchannel heat sinks but still outperforms the reference case, for a similar design complexity.

This work concerns detailed numerical modelling of injection fracturing in chalk formations under 2–3 km burial. The conditions and properties are chosen to resemble field conditions as typically found in oil fields subject to water injection in the Central Graben in the North Sea off-shore Denmark, Norway and the UK, but the studied phenomena are general to injection fracturing irrespective of geographical location and technological application. The complexity of reality is a challenging factor when mathematical models of hydraulic fractures in the subsurface are formulated. To this end the finite element method and a multi physics approach is instrumental, and data from fields developed and operated under conditions prudent for modelling are equally important. In the present study, a poroelastic finite element model is applied and used under plane strain conditions for a linear fracture mechanics investigation. This approach is suitable for analysing fracturing in arrays of parallel and horizontal wells as can be found in many oil fields undergoing water-flooding. Based on realistic field data, e.g. reservoir properties and rate and pressure scenarios representative for typical fields in the North Sea region, fracture initiation and fracture growth are analysed in details in a realistic field setting. Using a formulation of the J-integral, that includes Functionally Graded Material and poroelastic effects, it is demonstrated that formation fracture toughness determines fracture initiation and fracture height. The fracture propagation speed is controlled by continuity, e.g. the injection rate must balance the leak-off from the fracture wall. By combining the fields of geotechnical engineering, petroleum engineering and mechanical engineering with realistic data measurements, this study provides a novel realistic study of how the producer and injector pressure influence the fracturing process for induced hydraulic fractures in a line drive.

This paper presents a method for simultaneous optimization of the outer shape and internal topology of aircraft wings, with the objective of minimizing drag subject to lift and compliance constraints for multiple load cases. The physics are evaluated by the means of a source-doublet panel method for the aerodynamic response and linear elastic finite elements for the structural response, which are one way coupled. At each design iteration a mapping procedure is applied to map the current wing shape and corresponding pressure loads to the unfitted finite element mesh covering the design domain. Wings of small fixed-wing airplanes both, with and without a stiffening strut, are optimized. The resulting wings show internal topologies with struts and wall-truss combinations, depending on the design freedom of the shape optimization. The lift distributions of the optimized wings show patterns similar to the ones obtained when performing optimization of wing shapes with constraints on the bending moment at the root.

This work introduces an aeroelastic optimization framework with a coupled three-dimensional panel method and Timoshenko beam finite element model. The method allows for optimization of both the exterior surface of the wing and interior structural properties. We investigate the effects of curved wall spars on the aeroelastic performance of converged designs, which have been shown to provide an improved performance due to the ability to create tradeoffs between bending and torsional stiffnesses. Studies also highlight the importance of calculating aerodynamic loads in the deformed configuration and solving the coupled aeroelastic problem to convergence.

Variable thickness sheet and homogenization-based topology optimization often result in spread-out, non-well-defined solutions that are difficult to interpret or de-homogenize to sensible final designs. By extensive numerical investigations, we demonstrate that such solutions are due to non-uniqueness of solutions or at least very flat minima. Much clearer and better-defined solutions may be obtained by adding a measure of non-void space to the objective function with little if any increase in structural compliance. We discuss various alternatives for cleaning up solutions and propose two efficient approaches which both introduce an auxiliary field to control non-void space: one approach based on a cut element based auxiliary field (hybrid approach) and another approach based on an auxiliary element based field (density approach). At the end, we demonstrate significant qualitative and quantitative improvements in variable thickness sheet and de-homogenization designs resulting from the proposed cleaning schemes.

This work considers an efficient approach for length scale control in high-resolution three-dimensional level set–based topology optimization. A contrast parameter–based cut element method is employed to ensure a crisp interface representation while working on fixed background hexahedral meshes. This enables the use of efficient multigrid preconditioned Krylov methods in massively parallelized computations. The minimum length scale is controlled using a projection filter without the need for β-continuation. The capabilities of the proposed approach is demonstrated on several numerical examples using more than 62 million hexahedral elements.

Aircraft wings are commonly designed with nonplanar geometry, such as winglets, in order to improve aerodynamic efficiency. This work presents a method for generating nonplanar wing designs through gradient-based optimization, which is then used to investigate the performance characteristics of nonplanar wings. The nonplanar parameterization is defined to give a large design space that allows the formation of highly nonplanar features and permits large changes to the geometry. Aerodynamic characteristics are captured using an inviscid three-dimensional panel method with approximations for viscous drag. The methodology is demonstrated by optimizing reference wings from literature and comparing aerodynamic performance. Investigations are also performed on the impact on performance when wings are raised or drooped, and the differences in aerodynamic behavior between the two designs. Results suggest that the converged designs and their performance are highly dependent on how the geometry is restricted. If a large design space is provided, both raised and drooped wings are able to produce designs with similar performance when only inviscid analysis is considered. When accounting for viscous effects, results suggest that drooped wings are not beneficial for drag reduction.

A method for density-based topology optimization of heat exchangers with two fluids is proposed. The goal of the optimization process is to maximize the heat transfer from one fluid to the other, under maximum pressure drop constraints for each of the fluid flows. A single design variable is used to describe the physical fields. The solid interface and the fluid domains are generated using an erosion-dilation based identification technique, which guarantees well-separated fluids, as well as a minimum wall thickness between them. Under the assumption of laminar steady flow, the two fluids are modelled separately, but in the entire computational domain using the Brinkman penalization technique for ensuring negligible velocities outside of the respective fluid subdomains. The heat transfer is modelled using the convection-diffusion equation, where the convection is driven by both fluid flows. A stabilized finite element discretization is used to solve the governing equations. Results are presented for two different problems: a two-dimensional example illustrating and verifying the methodology; and a three-dimensional example inspired by shell-and-tube heat exchangers. The optimized designs for both cases show an improved heat transfer compared to the baseline designs. For the shell-and-tube case, the full freedom topology optimization approach is shown to yield performance improvements of up to 113% under the same pressure drop.

This review paper provides an overview of the literature for topology optimisation of fluid-based problems, starting with the seminal works on the subject and ending with a snapshot of the state of the art of this rapidly developing field. “Fluid-based problems” are defined as problems where at least one governing equation for fluid flow is solved and the fluid–solid interface is optimised. In addition to fluid flow, any number of additional physics can be solved, such as species transport, heat transfer and mechanics. The review covers 186 papers from 2003 up to and including January 2020, which are sorted into five main groups: pure fluid flow; species transport; conjugate heat transfer; fluid–structure interaction; microstructure and porous media. Each paper is very briefly introduced in chronological order of publication. A quantititive analysis is presented with statistics covering the development of the field and presenting the distribution over subgroups. Recommendations for focus areas of future research are made based on the extensive literature review, the quantitative analysis, as well as the authors’ personal experience and opinions. Since the vast majority of papers treat steady-state laminar pure fluid flow, with no recent major advancements, it is recommended that future research focuses on more complex problems, e.g., transient and turbulent flow.

This paper treats topology optimization of natural convection problems. A simplified model is suggested to describe the flow of a steady-state incompressible fluid, similar to Darcy's law for porous media. By neglecting inertia and viscous boundary layers, the flow model is significantly simplified. The fluid flow is coupled to the thermal convection-diffusion equation through the Boussinesq approximation. The coupled non-linear system of equations is discretized with stabilized finite elements and solved in a parallel framework that allows for the optimization of high-resolution three-dimensional problems. A density-based topology optimization approach is used, where the permeability and conductivity of the distributed material is interpolated. Due to the simplified model, the proposed methodology significantly reduces the computational effort required in the optimization. At the same time, it is notably more accurate than even simpler models that rely on Newton's law of cooling. The methodology discussed herein is applied to the optimization-based design of three-dimensional heat sinks. The final designs are compared with previous work obtained from solving the full set of Navier-Stokes equations, both in terms of design performance and computational cost. The computational time is shown to be decreased to around 5 20% in terms of core-hours, allowing for the possibility of optimizing designs during the workday on a small computational cluster and overnight on a high-end desktop. However, due to the use of a simplified model, the performance of the final designs are evaluated using the full Navier-Stokes equations. This ensures verification of performance, as well as systematic comparison with reference results.

The design of components for inertial microfluidic focusing and separation is primarily designed from basic geometric primitives. This paper presents a topology optimization methodology to the design synthesis of microfluidic particle manipulators. The flow is modeled by the Navier-Stokes equations in an Eulerian frame while the particle transport is modeled as a Lagrangian particle model. The model assumes that the particles are small and the suspension is dilute such that both the particle influence on the fluid motion and collisions between particles can be neglected. Two manipulator design problems are studied-one that makes particles follow a predefined trajectory, and another where particles are focused at the outlet utilizing the inertia of the particles. The latter is extracted and post-analyzed using a commercial software COMSOL verifying the method's ability to synthesize inertial microfluidic components.

The level set and density methods for topology optimization are often perceived as two very different approaches. This has to some extent led to two competing research directions working in parallel with only little overlap and knowledge exchange. In this paper, we conjecture that this is a misconception and that the overlap and similarities are far greater than the differences. To verify this claim, we employ, without significant modifications, many of the base ingredients from the density method to construct a crisp interface level set optimization approach using a simple cut element method. That is, we use the same design field representation, the same projection filters, the same optimizer, and the same so-called robust approach as used in density-based optimization for length scale control. The only noticeable difference lies in the finite element and sensitivity analysis, here based on a cut element method, which provides an accurate tool to model arbitrary, crisp interfaces on a structured mesh based on the thresholding of a level set—or density—field. The presented work includes a heuristic hole generation scheme and we demonstrate the design approach on several numerical examples covering compliance minimization and a compliant force inverter. Finally, we provide our MATLAB code, downloadable from www.topopt.dtu.dk, to facilitate further extension of the proposed method to, e.g., multiphysics problems.