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1. Design of bi-continuous structural battery electrolyte using a virtual temperature constrained topology optimization

   https://doi.org/10.1007/s00158-026-04253-6  

   Gorman, Jonathan; Black, Nolan; Pejman, Reza; Najafi, Ahmad R (February 2026, Structural and Multidisciplinary Optimization)

   Abstract Structural Battery Composites (SBC) are a new class of multifunctional materials that simultaneously provide structural load-bearing capabilities as well as electrochemical energy storage. By combining these functionalities, the weight and volume of energy storage systems can be reduced. In this work, a multiphysics, multi-objective topology optimization framework is used to design the Structural Battery Electrolyte (SBE) of the SBC. The objectives considered are the minimization of compliance and maximization of effective ionic conductivity. The multi-objective formulation is implemented using the Normalized-Normal Constraint method. This framework uses the finite element method to evaluate the structural and thermal responses and ResNet, a reduced order model, to find the effective ionic conductivity. The sensitivities of all physics are determined analytically using the adjoint sensitivity method. Previous studies have implemented topology optimization for designing SBE; however, these methodologies do not guarantee a bi-continuous design. Bi-continuous domains of the SBE are necessary for efficient ion transport. In this study, a virtual temperature constraint is used in conjunction with the topology optimization framework to prevent electrolyte islands from forming in the design domain. By doing so, bi-continuous behavior of the SBE microstructure can be enforced. Several numerical studies are conducted to demonstrate the capabilities of this framework. 

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2. Generative Design of Bionic Structures Via Concurrent Multiscale Topology Optimization and Conformal Geometry Method

   https://doi.org/10.1115/1.4047345  

   Jiang, Long; Gu, Xianfeng David; Chen, Shikui (January 2021, Journal of Mechanical Design)

   Abstract Topology optimization has been proved to be an efficient tool for structural design. In recent years, the focus of structural topology optimization has been shifting from single material continuum structures to multimaterial and multiscale structures. This paper aims at devising a numerical scheme for designing bionic structures by combining a two-stage parametric level set topology optimization with the conformal mapping method. At the first stage, the macro-structural topology and the effective material properties are optimized simultaneously. At the second stage, another structural topology optimization is carried out to identify the exact layout of the metamaterial at the mesoscale. The achieved structure and metamaterial designs are further synthesized to form a multiscale structure using conformal mapping, which mimics the bionic structures with “orderly chaos” features. In this research, a multi-control-point conformal mapping (MCM) based on Ricci flow is proposed. Compared with conventional conformal mapping with only four control points, the proposed MCM scheme can provide more flexibility and adaptivity in handling complex geometries. To make the effective mechanical properties of the metamaterials invariant after conformal mapping, a variable-thickness structure method is proposed. Three 2D numerical examples using MCM schemes are presented, and their results and performances are compared. The achieved multimaterial multiscale structure models are characterized by the “orderly chaos” features of bionic structures while possessing the desired performance. 

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3. Topology optimization with graph neural network enabled regularized thresholding

   https://doi.org/10.1016/j.eml.2024.102215  

   Gavris, Georgios Barkoulis; Sun, Waiching (September 2024, Extreme Mechanics Letters)

   Hsia, KJ; Rogers, JA; Suo, Z; Zhao, X (Ed.)

   Topology optimization algorithms often employ a smooth density function to implicitly represent geometries in a discretized domain. While this implicit representation offers great flexibility to parametrize the optimized geometry, it also leads to a transition region. Previous approaches, such as the Solid Isotropic Material Penalty (SIMP) method, have been proposed to modify the objective function aiming to converge toward integer density values and eliminate this non-physical transition region. However, the iterative nature of topology optimization renders this process computationally demanding, emphasizing the importance of achieving fast convergence. Accelerating convergence without significantly compromising the final solution can be challenging. In this work, we introduce a machine learning approach that leverages the message-passing Graph Neural Network (GNN) to eliminate the non-physical transition zone for the topology optimization problems. By representing the optimized structures as weighted graphs, we introduce a generalized filtering algorithm based on the topology of the spatial discretization. As such, the resultant algorithm can be applied to two- and three-dimensional space for both Cartesian (structured grid) and non-Cartesian discretizations (e.g. polygon finite element). The numerical experiments indicate that applying this filter throughout the optimization process may avoid excessive iterations and enable a more efficient optimization procedure. 

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4. Conforming virtual element method for nondivergence form linear elliptic equations with Cordes coefficients

   https://doi.org/10.1142/S0218202525500034  

   Bonnet, Guillaume; Cangiani, Andrea; Nochetto, Ricardo H (January 2025, Mathematical Models and Methods in Applied Sciences)

   We propose and analyze an [Formula: see text]-conforming virtual element method (VEM) for the simplest linear elliptic PDEs in nondivergence form with Cordes coefficients. The VEM hinges on a hierarchical construction valid for any dimension [Formula: see text]. The analysis relies on the continuous Miranda–Talenti estimate for convex domains [Formula: see text] and is rather elementary. We prove stability and error estimates in [Formula: see text], including the effect of quadrature, under minimal regularity of the data. Numerical experiments illustrate the interplay of coefficient regularity and convergence rates in [Formula: see text]. 

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5. Thermal Conductivity Modeling and Optimization of Ceramic Matrix Composite Spinodoids

   https://doi.org/10.2514/6.2026-0221  

   Yildiz, Saltuk; Acar, Pinar (January 2026, American Institute of Aeronautics and Astronautics)

   Spinodoids exhibit unique non-periodic architectures that offer distinct advantages in mechanical performance compared to conventional architected materials such as triply periodic minimal surfaces (TPMS) and lattice structures. In this study, spinodal topologies are optimized to enhance thermal conductivity, particularly for use in high-temperature aerospace applications. For this purpose, two-dimensional (2D) dual-phase spinodal structures are first generated and mapped into three-dimensional (3D) geometries, and their anisotropic thermal conductivities are evaluated along different spatial directions. A mathematical optimization framework is then implemented to perform topology optimization of spinodoids having a base material of SiC/SiC ceramic matrix composite (CMC). The spinodal structures are parameterized through orientation distributions constrained by conical angles. The steady-state thermal conduction is simulated numerically to compute elemental heat fluxes in three Cartesian directions, and effective thermal conductivities are extracted via the volume-averaging method. The optimization problem defines a composite objective function based on the anisotropic effective conductivities and is solved using a gradient-free heuristic algorithm. The resulting optimum design demonstrates the potential of spinodoid structures to improve thermal management in hot-section components of gas turbine engines. 

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