Topology optimization (TO) holds the promise of designing next-generation compact and efficient fluidic devices. However, the inherent complexity of fluid-based TO systems, characterized by multiphysics nonlinear interactions, poses substantial barriers to entry for researchers. Beyond the inherent intricacies of forward simulation models, design optimization is further complicated by the difficulty of computing sensitivities, i.e., gradients. Manual derivation and implementation of sensitivities are often laborious and prone to errors, particularly for non-trivial objectives, constraints, and material models. An alternative solution is automatic differentiation (AD). Although AD has been previously demonstrated for simpler TO problems, extending its use to complex nonlinear multiphysics systems, specifically in fluidic optimization, is key to reducing the entry barrier. To this end, we introduce TOFLUX, a TO framework for fluid devices leveraging the JAX library for high-performance automatic differentiation. The flexibility afforded by AD enables the rapid exploration and evaluation of various objectives and constraints. We illustrate this capability through challenging examples encompassing thermo-fluidic coupling, fluid–structure interaction, and non-Newtonian flows. Additionally, we demonstrate the seamless integration of our framework with neural networks and machine learning methodologies, enabling modern approaches to scientific computing. Ultimately, the framework aims to provide a foundational resource to accelerate research and innovation in fluid-based TO. The software accompanying this educational paper can be accessed at https://github.com/UW-ERSL/TOFLUX .

Latent heat thermal energy storage (LHTES) systems are compelling candidates for energy storage, primarily owing to their high storage density. Improving their performance is crucial for developing the next-generation efficient and cost effective devices. Topology optimization (TO) has emerged as a powerful computational tool to design LHTES systems by optimally distributing a high-conductivity material (HCM) and a phase change material (PCM). However, conventional TO typically limits itself to optimizing the geometry for fixed, pre-selected materials. This approach does not leverage the large and expanding databases of novel materials. Consequently, the co-design of material and geometry for LHTES remains a challenge and is largely unexplored.To address this limitation, we present an automated design framework for the concurrent optimization of material choice and topology. A key challenge is the discrete nature of material selection, which is incompatible with the gradient-based methods used for TO. We overcome this by using a data-driven variational autoencoder (VAE) to project discrete material databases for both the HCM and PCM onto continuous and differentiable latent spaces. These continuous material representations are integrated into an end-to-end differentiable, transient nonlinear finite-element solver that accounts for phase change. We demonstrate this framework on a problem aimed at maximizing the discharged energy within a specified time, subject to cost constraints. The effectiveness of the proposed method is validated through several illustrative examples.

Quantum Singular Value Transformation (QSVT) provides a unified framework for applying polynomial functions to the singular values of a block-encoded matrix. QSVT prepares a state proportional to $\bA^{-1}\bb$ with circuit depth $O(d\cdot\mathrm{polylog}(N))$, where $d$ is the polynomial degree of the $1/x$ approximation and $N$ is the size of $\bA$. Current polynomial approximation methods are over the continuous interval $[a,1]$, giving $d = O(\sqrt{\kap}\log(1/\varepsilon))$, and make no use of any properties of $\bA$. We observe here that QSVT solution accuracy depends only on the polynomial accuracy at the eigenvalues of $\bA$. When all $N$ eigenvalues are known exactly, a pure spectral polynomial $p_{S}$ can interpolate $1/x$ at these eigenvalues and achieve unit fidelity at reduced degree. But its practical applicability is limited. To address this, we propose a spectral correction that exploits prior knowledge of $K$ eigenvalues of $\bA$. Given any base polynomial $p_0$, such as Remez, of degree $d_0$, a $K\times K$ linear system enforces exact interpolation of $1/x$ only at these $K$ eigenvalues without increasing $d_0$. The spectrally corrected polynomial $p_{SC}$ preserves the continuous error profile between eigenvalues and inherits the parity of $p_0$. QSVT experiments on the 1D Poisson equation demonstrate up to a $5\times$ reduction in circuit depth relative to the base polynomial, at unit fidelity and improved compliance error. The correction is agnostic to the choice of base polynomial and robust to eigenvalue perturbations up to $10\%$ relative error. Extension to the 2D Poisson equation suggests that correcting a small fraction of the spectrum may suffice to achieve fidelity above $0.999$.

Computed Tomography Angiography (CTA) is widely used to reconstruct vascular geometry from projection measurements, with conventional approaches such as Filtered Back-Projection (FBP) and Iterative Reconstruction (IR) forming the clinical standard. Blood flow is subsequently estimated through Computational Fluid Dynamics (CFD) simulations, which require vascular geometry and boundary conditions to be specified a priori. Since the geometry is fixed prior to flow estimation, the recovery of unknown anatomical features (e.g., missing branches or stenoses) is precluded. In this work, we present a fluid-physics-constrained reconstruction framework that leverages topology optimization (TO) to jointly recover vascular geometry and blood velocity directly from time-resolved CTA sinograms. The formulation couples a steady incompressible flow model with a transient advection-diffusion contrast transport model, mapped to sinogram space through a differentiable projection operator. The recovered velocity fields provide hemodynamic information and can support downstream estimation of wall shear stress and flow distribution, without requiring a separate CFD pipeline. The proposed method is demonstrated on synthetic phantoms under varying sparsity and noise levels, and on representative projection data.

This paper presents STORX: Shape and Topology Optimization for Research and Experimentation, an open-source MATLAB-based educational framework for learning and teaching computational design optimization. STORX provides a platform for parametric and level-set shape optimization, as well as topology optimization methods including density, level-set, and topological sensitivity approaches such as evolutionary and Pareto-tracing methods. All modules follow a consistent object-oriented structure and integrate visualization, sensitivity analysis, and finite element routines, enabling users to explore the continuum between shape and topology optimization in a transparent and reproducible manner. The code is designed to complement graduate-level coursework and independent research by emphasizing modularity and extensibility through a clear separation of intent. Core software interfaces are defined via abstract base classes, enabling new objective functionals and design/manufacturing constraints to be implemented by adding derived classes without modifying the core code. The paper also describes the software architecture and demonstrates how the framework maps mathematical formulations directly to executable code through a series of illustrative problems.

Optimizing lattice structures for energy absorption and load-bearing applications necessitates accurately capturing their nonlinear mechanical response under large deformation. However, traditional nonlinear finite element analysis (NL-FEA) can often fail, particularly at higher compression, which creates numerical gaps in the design space, hindering gradient-based optimization algorithms, ultimately resulting in suboptimal designs. We propose here a neural network (NN) surrogate model that is trained on data generated using Abaqus, a commercial FEA solver. A unique feature of the model is that it exploits both fully and partially successful NL-FEA simulations. This significantly improves the surrogate model’s accuracy and predictive coverage across the design space. The resulting model enables efficient, gradient-driven optimization of lattice structures to match a desired force-displacement response. By replacing expensive NL-FEA evaluations, the surrogate model substantially reduces computational cost while maintaining a coefficient of determination ( R^2 ) of 0.98 in prediction. The effectiveness and versatility of this framework are demonstrated by successfully optimizing 3 × 3 lattice structures under 40

Abstract Quantum algorithms require encoding classical vectors as quantum states, a step known as amplitude encoding. General-purpose routines produce circuits with O ( 2 m ) gates for vectors of length N = 2 m , for an m -qubit register. However, vectors arising in scientific and engineering applications often exhibit mathematical structure that admits far more efficient encoding. Theoretical work over the last decade has established efficient circuits for several structured vector classes, but without open-source implementations. We present PyEncode , an open-source Python library that implements this body of theory in a unified framework. It covers ten exact pattern families: sparse, step, square, Walsh, Fourier, geometric, Hamming, staircase, Dicke , and polynomial . A function encode maps each pattern to a verified Qiskit circuit, with no vector materialization and no approximation; for example, encode(SPARSE([(19, 1.0)]), N = 64) encodes the vector e 19 of length N = 64 . Sparse, step, Walsh, Hamming, and staircase patterns require O ( m ) gates; square and Fourier patterns require O ( m 2 ) ; Dicke states | D k m ⟩ require O ( k ( m − k ) ) , that denotes uniform superpositions over indices of Hamming weight k ; degree- d polynomials require O ( m d + 1 ) . A companion predict_gates function estimates transpiled gate counts without synthesis, and a reverse-lookup utility match_vector identifies which family best fits a given numerical vector. Three composition primitives are supported: SUM for weighted superpositions, PARTITION for ancilla-free composition of disjoint-support patterns, and TENSOR for separable states over disjoint subregisters. For amplitude vectors outside these exact families, PyEncode also provides a matrix product state (MPS) loader, encode_mps for approximate vector encoding. The library is available at https://github.com/UW-ERSL/PyEncode .

The Finite element method (FEM) has long served as the computational backbone for topology optimization (TO). However, for designing structures undergoing large deformations, conventional FEM-based TO often exhibits numerical instabilities due to severe mesh distortions, tangling, and large rotations, consequently leading to convergence failures. To address this challenge, we present a TO framework based on the Material Point Method (MPM). MPM is a hybrid Lagrangian-Eulerian particle method, well-suited for simulating large deformations. In particular, we present an end-to-end differentiable implicit MPM framework for designing structures undergoing quasi-static hyperelastic large deformations. The effectiveness of the approach is demonstrated through validation studies encompassing both single and multi-material designs, including the design of compliant soft robotic grippers. The software accompanying this paper can be accessed at github.com/UW-ERSL/MOTO.

Quantum algorithms require encoding classical vectors as quantum states, a step known as amplitude encoding. General-purpose state preparation routines accept any input vector of length N = 2^m and produce circuits with 2^m gates. However, vectors arising in scientific and engineering applications often exhibit mathematical structure that admits far more efficient encoding. Recent theoretical work has established closed-form circuits for several structured vector classes, but without open-source implementations. We present PyEncode, an open-source Python library that implements this body of theory in a unified, immediately deployable framework. The library covers sparse, step, square (general interval), Walsh, geometric, and Fourier patterns, and supports weighted superpositions of pattern states via the linear combination of unitaries (LCU) protocol, enabling exact preparation of piecewise-structured vectors such as multi-interval Hamiltonians. PyEncode exposes a single function encode(VectorObj, N) that maps a typed parameter declaration directly to a verified Qiskit circuit, with no vector materialization and no approximation. Sparse, step, and Walsh vectors require only m gates; geometric (exponential-decay) vectors require m gates with zero two-qubit gates; square (general interval) vectors require m^2 gates via a QFT-based constant adder, with m special cases; Fourier (sinusoidal) vectors require m^2 gates via the inverse Quantum Fourier Transform – all exponentially fewer than the 2^m cost of general state preparation. LCU combines r component circuits whose total gate cost is the sum of individual component costs, with success probability p ∈ (0,1] determined analytically. The library is available at https://github.com/UW-ERSL/PyEncode.

This work presents a novel approach for designing multi-alloy structures by simultaneously optimizing the material and topology. The proposed computational framework employs a material latent space integrated with density-based topology optimization in a two-step process. In the first step, a variational autoencoder (VAE), a type of neural network, maps a database of materials and their properties to a continuous, low-dimensional latent space. In the second step, the latent space is coupled with topology design variables (pseudo-densities) to simultaneously optimize material and topology. A gradient-based optimizer, specifically the method of moving asymptotes (MMA), traverses the latent space to select the material while simultaneously optimizing the topology. Optionally, one can add a penalization term that will drive the latent points towards real materials from the dataset. The framework is illustrated through 2D and 3D numerical examples using up to 20 materials, involving more than a million degrees of freedom.

Many engineering problems involve solving large linear systems of equations. Conjugate gradient (CG) is one of the most popular iterative methods for solving such systems. However, CG typically requires a good preconditioner to speed up convergence. One such preconditioner is the sparse approximate inverse (SPAI). In this paper, we explore the computation of an SPAI on quantum annealing machines by solving a series of quadratic unconstrained binary optimization (QUBO) problems. Numerical experiments are conducted using both well-conditioned and poorly-conditioned linear systems arising from a 2D finite difference formulation of the Poisson problem.

Feature-mapping methods for topology optimization (FMTO) facilitate direct geometry extraction by leveraging high-level geometric descriptions of the designs. However, FMTO often relies solely on Boolean unions, which can restrict the design space. This work proposes an FMTO framework leveraging an expanded set of Boolean operations, namely, union, intersection, and subtraction. The optimization process entails determining the primitives and the optimal Boolean operation tree. In particular, the framework leverages a recently proposed unified Boolean operation approach. This approach presents a continuous and differentiable function that interpolates the Boolean operations, enabling gradient-based optimization. The proposed methodology is agnostic to the specific primitive parametrization and is showcased through various numerical examples.

Catalogs have been used for over a century for designing engineering systems. While catalogs are excellent repositories of engineering information, they are difficult to navigate, specifically to spot clusters, gaps, substitutes, and outliers. Inspired by Ashby charts for material selection, we propose here, a visual representation of engineering catalogs using neural networks. In particular, we employ variational autoencoders (VAEs) to project catalog data onto a lower-dimensional latent space. The latent space can then be visualized to explore the underlying structure of the catalog. Specifically, creators can use this visual representation to identify gaps and outliers in their data, while end users can benefit from this representation to compare catalogs from competitors, and to find substitutes. Contours can be super-imposed on the charts to enable selection based on user-defined attributes; these contours are generalization of design indices associated with Ashby charts. Various examples of catalogs across engineering disciplines, ranging from materials and bearings to motors and batteries are illustrated using the proposed method. Using these examples, we (1) study the impact of the latent space dimension on the representational error, (2) illustrate how designers can easily choose alternate configurations based on their design requirements, and (3) gaps in catalog offerings can be clearly identified, providing a stimulus for new product development.

In an increasingly competitive and digital industrial environment, the optimization of structures is a key point not only to reduce costs but also to reduce the consumption of natural resources. To this end, different approaches have emerged throughout history based on the tools available at the time. With the current rise of artificial intelligence and the concept of machine learning, revolutionary ideas are emerging that allow an optimal dimensioning of structures in record time. This work presents the use of variational autoencoders and mixed-variable solvers as a proposal for structural optimization and material selection. It has expanded upon previous research by advancing in three directions: (1) incorporating more material attributes, particularly relevant for environmental considerations; (2) analyzing in more detail aspects of VAEs such as the dimensionality of the latent space; and (3) a two-step hybrid approach to select the optimal candidate: preliminary filtering with VAE and final design via mixed-variable model. Various examples demonstrate the applicability of the proposed method.

Support structures act as primary conduits for heat flow in laser powder bed fusion (LPBF). Frame supports, as opposed to block-type supports, have proven to be a good choice for LPBF, with no metal powder entrapment and ease of removal. In this paper, we use a multi-load formulation, where different loads at different instances of time are used for frame support optimization, to constrain the structural deformation of the part during printing. Parts and supports are analyzed in tandem at the end of each layer build, to account for the role of support stiffness in the overall part deformation. The proposed framework prevents recoater collision, and controls the geometric accuracy of the part, by optimizing the size of frame supports. Numerical results for the optimal volume of manufacturable frame support are reported for geometries with varying overhang characteristics.

Tangled (non-convex) elements, i.e. elements with negative Jacobian determinant, can lead to erroneous results in the standard finite element method (FEM). Constructing tangle-free, well-structured meshes for complex geometries is often impossible. Hence there is a need to explore analysis methods that can directly handle such tangled meshes. In this paper, we propose the isoparametric tangled finite element method (i-TFEM) for free and forced vibration problems over tangled meshes. By employing piece-wise invertible mapping, a variational formulation is derived, leading to a simple modification of the standard FEM stiffness and mass matrices with the incorporation of additional compatibility constraints. Moreover, i-TFEM reduces to standard FEM for non-tangled (regular) meshes. The proposed method is implemented for three types of elements: 4-node quadrilateral, 9-node quadrilateral, and 8-node hexahedral elements. The numerical results demonstrate that i-TFEM is able to consistently handle general tangled (non-convex) elements, enabling convenient meshing for complex geometries.

In this paper, we present a framework for multiscale topology optimization of fluid-flow devices. The objective is to minimize dissipated power, subject to a desired contact-area. The proposed strategy is to design optimal microstructures in individual finite element cells, while simultaneously optimizing the overall fluid flow. In particular, parameterized super-shape microstructures are chosen here to represent microstructures since they exhibit a wide range of permeability and contact area. To avoid repeated homogenization, a finite set of these super-shapes are analyzed a priori, and a variational autoencoder (VAE) is trained on their fluid constitutive properties (permeability), contact area and shape parameters. The resulting differentiable latent space is integrated with a coordinate neural network to carry out a global multi-scale fluid flow optimization. The latent space enables the use of new microstructures that were not present in the original data-set. The proposed method is illustrated using numerous examples in 2D.

Solving linear systems of equations is an important problem in engineering. Many quantum algorithms, such as the Harrow-Hassidim-Lloyd algorithm and the box algorithm, have been proposed for solving such systems. The focus of this article is on improving the efficiency of the box algorithm. The basic principle behind this algorithm is to transform the linear system into a series of quadratic unconstrained binary optimization (QUBO) problems, which are then solved on annealing machines. The computational efficiency of the box algorithm is entirely determined by the number of iterations, which, in turn, depends on the box contraction ratio, typically set to 0.5. Here, it is shown through theoretical analysis that a contraction ratio of 0.5 is sub-optimal and that a computational speed-up can be achieved with a contraction ratio of 0.2. This is confirmed through numerical experiments where a computational speed-up between $ 20 \% $ 20% to $ 60 \% $ 60% is observed when the optimal contraction ratio is used.

Cellular structures found in nature exhibit remarkable properties such as high strength, high energy absorption, excellent thermal/acoustic insulation, and fluid transfusion. Many of these structures are Voronoi-like; therefore researchers have proposed Voronoi multi-scale designs for a wide variety of engineering applications. However, designing such structures can be computationally prohibitive due to the multi-scale nature of the underlying analysis and optimization. In this work, we propose the use of a neural network (NN) to carry out efficient topology optimization (TO) of multi-scale Voronoi structures. The NN is first trained using Voronoi parameters (cell site locations, thickness, orientation, and anisotropy) to predict the homogenized constitutive properties. This network is then integrated into a conventional TO framework to minimize structural compliance subject to a volume constraint. Special considerations are given for ensuring positive definiteness of the constitutive matrix and promoting macroscale connectivity. Several numerical examples are provided to showcase the proposed method.