This paper presents an efficient and easy-to-implement method for imposing feature size in level set-based topology optimization. A minimum length scale is enforced via a penalty based on a spread skeleton that captures the location of the skeleton and the required distance to it. The skeleton is constructed using first-order gradient information by solving a heat conduction problem and smoothing the resulting gradient with a PDE-based filter. Unlike other approaches, the method does not require to carry out integration over the skeleton, construct additional integration domains, or build an accurate distance from the skeleton. The work focuses on the generation of spread skeletons and the influence of the formulated penalty on optimization results. The design geometry is represented by a level set function, and structural responses are predicted using the extended finite element method. Optimization problems are solved using gradient-based algorithms, and the sensitivity analysis is performed with the adjoint approach. The ability of the proposed method to accurately control length scale is demonstrated with compliance minimization problems under a volume constraint. Similarly to other feature size control schemes, the developed geometric penalty tends to inhibit topological changes, especially in two dimensions, and results in a high dependence on the initial layout. An activation strategy that successively recruits regions of the design domain based on the spread skeleton value is used to avoid early convergence to suboptimal solutions. Numerical examples demonstrate the potential of the proposed framework to generate designs exhibiting both enhanced performance and a minimum length scale.

This paper presents for the first time an adaptive immersed approach for level-set topology optimization using higher-order truncated hierarchical B-spline discretizations for design and state variable fields. Boundaries and interfaces are represented implicitly by the iso-contour of one or multiple level-set functions. An immersed finite element method, the eXtended IsoGeometric Analysis, is used to predict the physical response. The proposed optimization framework affords different adaptively refined higher-order B-spline discretizations for individual design and state variable fields. The increased continuity of higher-order B-spline discretizations together with local refinement enables direct control over the accuracy of the representation of each field while simultaneously reducing computational cost compared to uniformly refined discretizations. A flexible mesh adaptation strategy enables local refinement based on geometric measures or physics-based error indicators. These adaptive discretization and analysis approaches are integrated into gradient-based optimization schemes, evaluating the design sensitivities using the adjoint method. Numerical studies illustrate the features of the proposed framework with static, linear elastic, multi-material, two- and three-dimensional problems. The examples provide insight into the effect of refining the design variable field on the optimization result and the convergence rate of the optimization process. Using coarse higher-order B-spline discretizations for level-set fields promotes the development of smooth designs and suppresses the emergence of small features. Moreover, adaptive mesh refinement for state variable fields results in a reduction of overall computational cost. Higher-order B-spline discretizations are especially interesting when evaluating gradients of state variable fields due to their higher inter-element continuity.

Immersed finite element methods provide a convenient analysis framework for problems involving geometrically complex domains, such as those found in topology optimization and microstructures for engineered materials. However, their implementation remains a major challenge due to, among other things, the need to apply nontrivial stabilization schemes and generate custom quadrature rules. This article introduces the robust and computationally efficient algorithms and data structures comprising an immersed finite element preprocessing framework. The input to the preprocessor consists of a background mesh and one or more geometries defined on its domain. The output is structured into groups of elements with custom quadrature rules formatted such that common finite element assembly routines may be used without or with only minimal modifications. The key to the preprocessing framework is the construction of material topology information, concurrently with the generation of a quadrature rule, which is then used to perform enrichment and generate stabilization rules. While the algorithmic framework applies to a wide range of immersed finite element methods using different types of meshes, integration, and stabilization schemes, the preprocessor is presented within the context of the extended isogeometric analysis. This method utilizes a structured B-spline mesh, a generalized Heaviside enrichment strategy considering the material layout within individual basis functions’ supports, and face-oriented ghost stabilization. Using a set of examples, the effectiveness of the enrichment and stabilization strategies is demonstrated alongside the preprocessor’s robustness in geometric edge cases. Additionally, the performance and parallel scalability of the implementation are evaluated.

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.

Solving conjugate heat transfer design problems is relevant for various engineering applications requiring efficient thermal management. Heat exchange between fluid and solid can be enhanced by optimizing the system layout and the shape of the flow channels. As heat is transferred at fluid/solid interfaces, it is crucial to accurately resolve the geometry and the physics responses across these interfaces. To address this challenge, this work investigates for the first time the use of an eXtended Finite Element Method (XFEM) approach to predict the physical responses of conjugate heat transfer problems considering turbulent flow. This analysis approach is integrated into a level set-based optimization framework. The design domain is immersed into a background mesh and the geometry of fluid/solid interfaces is defined implicitly by one or multiple level set functions. The level set functions are discretized by higher-order B-splines. The flow is predicted by the Reynolds Averaged Navier–Stokes equations. Turbulence is described by the Spalart–Allmaras model and the thermal energy transport by an advection–diffusion model. Finite element approximations are augmented by a generalized Heaviside enrichment strategy with the state fields being approximated by linear basis functions. Boundary and interface conditions are enforced weakly with Nitsche’s method, and the face-oriented ghost stabilization is used to mitigate numerical instabilities associated with the emergence of small integration subdomains. The proposed XFEM approach for turbulent conjugate heat transfer is validated against benchmark problems. Optimization problems are solved by gradient-based algorithms and the required sensitivity analysis is performed by the adjoint method. The proposed framework is illustrated with the design of turbulent heat exchangers in two dimensions. The optimization results show that, by tuning the shape of the fluid/solid interface to generate turbulence within the heat exchanger, the transfer of thermal energy can be increased.

This paper presents an immersed, isogeometric finite element framework to predict the response of multi-material, multi-physics problems with complex geometries using locally refined discretizations. To circumvent the need to generate conformal meshes, this work uses an extended finite element method (XFEM) to discretize the governing equations on non-conforming, embedding meshes. A flexible approach to create truncated hierarchical B-splines discretizations is presented. This approach enables the refinement of each state variable field individually to meet field-specific accuracy requirements. To obtain an immersed geometry representation that is consistent across all hierarchically refined B-spline discretizations, the geometry is immersed into a single mesh, the XFEM background mesh, which is constructed from the union of all hierarchical B-spline meshes. An extraction operator is introduced to represent the truncated hierarchical B-spline bases in terms of Lagrange shape functions on the XFEM background mesh without loss of accuracy. The truncated hierarchical B-spline bases are enriched using a generalized Heaviside enrichment strategy to accommodate small geometric features and multi-material problems. The governing equations are augmented by a formulation of the face-oriented ghost stabilization enhanced for locally refined B-spline bases. We present examples for two- and three-dimensional linear elastic and thermo-elastic problems. The numerical results validate the accuracy of our framework. The results also demonstrate the applicability of the proposed framework to large, geometrically complex problems.

Multi-material problems often exhibit complex geometries along with physical responses presenting large spatial gradients or discontinuities. In these cases, providing high-quality body-fitted finite element analysis meshes and obtaining accurate solutions remain challenging. Immersed boundary techniques provide elegant solutions for such problems. Enrichment methods alleviate the need for generating conforming analysis grids by capturing discontinuities within mesh elements. Additionally, increased accuracy of physical responses and geometry description can be achieved with higher-order approximation bases. In particular, using B-splines has become popular with the development of IsoGeometric Analysis. In this work, an eXtended IsoGeometric Analysis (XIGA) approach is proposed for multi-material problems. The computational domain geometry is described implicitly by level set functions. A novel generalized Heaviside enrichment strategy is employed to accommodate an arbitrary number of materials without artificially stiffening the physical response. Higher-order B-spline functions are used for both geometry representation and analysis. Boundary and interface conditions are enforced weakly via Nitsche’s method, and a new face-oriented ghost stabilization methodology is used to mitigate numerical instabilities arising from small material integration subdomains. Two- and three-dimensional heat transfer and elasticity problems are solved to validate the approach. Numerical studies provide insight into the ability to handle multiple materials considering sharp-edged and curved interfaces, as well as the impact of higher-order bases and stabilization on the solution accuracy and conditioning.