arXiv subjects

Shengfeng Zhu

Publications and source records attributed to Shengfeng Zhu.

16 recordsLinked to original sources

arXiv2026

Adaptive Time Windows for Discrete Adjoint Topology Optimization of Unsteady Flows

Rather than prescribing an evaluation interval a priori, the proposed framework characterizes each evolving unsteady flow using a sequence of time windows. A consecutive-window convergence criterion is introduced to automatically identify a representative time window within its fully developed stage. The objective evaluation and discrete adjoint analysis are then carried out consistently over the identified representative time window. The framework is implemented using a regularized lattice Boltzmann method-based large-eddy simulation (LBM-LES) solver together with a partial bounce-back fluid-solid model. The consecutive-window convergence criterion is first validated using the backward-facing step flow. Cylinder-flow applications are then employed to investigate the influence of different flow regimes on the proposed framework. The wake-flow recovery problem verifies its effectiveness for unsteady topology optimization, while U-bend optimization further demonstrates its capability to identify and reorganize complex vortical structures.

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arXiv2026

A Unified Phase-Field Fourier Neural Network Framework for Topology Optimization

We propose Alternating Phase-Field Fourier Neural Networks (APF-FNNs) as a unified and physics-based framework for topology optimization. The approach decouples the design problem by representing the state, adjoint, and topology fields with three separate Fourier neural networks, which are trained via a stable collaborative alternating scheme applicable to both self-adjoint and non-self-adjoint problems. To obtain well-resolved designs, the Ginzburg--Landau energy functional is embedded in the loss of the topology network as an intrinsic regularizer, naturally enforcing smooth and distinct interfaces between the two phases. Phase-field updates are driven by adjoint-based optimality conditions, and design sensitivities are evaluated efficiently using automatic differentiation, ensuring that the gradients correspond to exact total derivatives rather than naive partial derivatives. In contrast to classical phase-field methods, APF-FNNs exploit these physically consistent design gradients directly, avoiding pseudo-time gradient-flow solvers. By formulating physics-driven losses from variational principles or strong-form PDE residuals, the framework is broadly applicable to 2D and 3D benchmark problems, including compliance minimization, eigenvalue maximization, and Stokes/Navier--Stokes flow optimization. Across these examples, APF-FNNs consistently yield competitive performance and well-resolved topologies, establishing a versatile and scalable foundation for physics-driven computational design.

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arXiv2025

Shape design with phase field methods for structural hemivariational inequalities in contact problems

We develop mathematical models for shape design and topology optimization in structural contact problems involving friction between elastic and rigid bodies. The governing mechanical constraint is a nonlinear, non-smooth, and non-convex hemivariational inequality, which provides a more general and realistic description of frictional contact forces than standard variational inequalities, but is also more challenging due to its non-convexity. For energy-type shape functionals, the Eulerian derivative of the hemivariational inequality is derived through rigorous shape sensitivity analysis. The rationality of a regularization approach is justified by asymptotic analysis, and this method is further applied to handle the non-smoothness of general shape functionals in the sensitivity framework. Based on these theoretical results, a numerical boundary variational method is proposed for shape optimization. For topology optimization, three phase-field algorithms are developed: a gradient-flow phase-field method, a phase-field method with second-order regularization of the cost functional, and a phase-field method coupled with topological derivatives. To the best of our knowledge, these approaches are new for shape design in hemivariational inequalities. Various numerical experiments confirm the accuracy and effectiveness of the proposed shape and topology optimization algorithms.

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arXiv2025

Maximization of Supercapacitor Storage via Topology Optimization of Electrode Structures

As widely used electrochemical storage devices, supercapacitors deliver higher power density than batteries, but suffer from significantly lower energy density. In this work, we propose a topology optimization model for electrode structure to maximize energy storage in supercapacitors. The existence of minimizers to the resulting optimal control problem, which is constrained by a modified steady-state Poisson--Nernst--Planck system describing ionic electrodiffusion, has been theoretically established by using the direct method in the calculus of variation. Sensitivity analysis of the topology optimization model is performed to derive variational derivatives and corresponding adjoint equations. A gradient flow formulation discretized by a stabilized semi-implicit scheme is developed to solve the resulting topology optimization problem. Extensive numerical experiments present various porous electrode structures that own large area of electrode-electrolyte interface, demonstrating the effectiveness and robustness of the proposed topology optimization model and corresponding algorithm.

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arXiv2025

Bioluminescence tomography via a shape optimization method based on a complex-valued model

In this study, we investigate the inverse source problem arising in bioluminescence tomography, the objective of which is to reconstruct both the support and the intensity of an internal light source from boundary measurements governed by an elliptic model. A shape optimization framework is developed in which the source intensity and its support are decoupled through first-order optimality conditions. To enhance the stability of the reconstruction, we incorporate a parameter-dependent coupled complex boundary method together with perimeter and volume regularizations. Source support is represented by a level set function, allowing the algorithm to naturally accommodate topological changes and recover multiple, closely spaced, or nested source regions. Theoretical justifications for the proposed formulation and regularization strategy are established, and extensive numerical experiments are performed to assess the reconstruction accuracy for both noise-free and noisy data. The results demonstrate that our method achieves robust and accurate recovery of source geometry and intensity, and exhibits clear advantages over existing approaches.

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arXiv2025

Topology optimization for microfluidic mixers by a phase field method

We investigate multi-physical topology optimization for microfluidic mixers employing the phase-field model. The optimization problem is formulated using a modified Ginzburg-Landau free energy functional. To eliminate fluid blockage in microfluidic mixers, we incorporate the coupled Navier-Stokes, convection-diffusion and Poisson-Boltzmann equations. An Allen-Cahn type gradient flow method is proposed based on sensitivity analysis. The algorithm is validated for its computational effectiveness through numerical simulations of benchmark problems in 2D and 3D.

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arXiv2025

Convergence Analysis of an Adaptive Nonconforming FEM for Phase-Field Dependent Topology Optimization in Stokes Flow

In this work, we develop an adaptive nonconforming finite element algorithm for the numerical approximation of phase-field parameterized topology optimization governed by the Stokes system. We employ the conforming linear finite element space to approximate the phase field, and the nonconforming linear finite elements (Crouzeix-Raviart elements) and piecewise constants to approximate the velocity field and the pressure field, respectively. We establish the convergence of the adaptive method, i.e., the sequence of minimizers contains a subsequence that converges to a solution of the first-order optimality system, and the associated subsequence of discrete pressure fields also converges. The analysis relies crucially on a new discrete compactness result of nonconforming linear finite elements over a sequence of adaptively generated meshes. We present numerical results for several examples to illustrate the performance of the algorithm, including a comparison with the uniform refinement strategy.

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arXiv2025

Shape Optimization of Supercapacitor Electrode to Maximize Charge Storage

We build a new mathematical model of shape optimization for maximizing ionic concentration governed by the multi-physical coupling steady-state Poisson-Nernst-Planck system. Shape sensitivity analysis is performed to obtain the Eulerian derivative of the cost functional. The Gummel fixed-point method with inverse harmonic averaging technique on exponential coefficient is used to solve efficiently the steady-state Poisson-Nernst-Planck system. Various numerical results using a shape gradient algorithm in 2d and 3d are presented.

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arXiv2025

On the Crouzeix-Raviart Finite Element Approximation of Phase-Field Dependent Topology Optimization in Stokes Flow

In this work, we investigate a nonconforming finite element approximation of phase-field parameterized topology optimization governed by the Stokes flow. The phase field, the velocity field and the pressure field are approximated by conforming linear finite elements, nonconforming linear finite elements (Crouzeix-Raviart elements) and piecewise constants, respectively. When compared with the standard conforming counterpart, the nonconforming FEM can provide an approximation with fewer degrees of freedom, leading to improved computational efficiency. We establish the convergence of the resulting numerical scheme in the sense that the sequences of phase-field functions and discrete velocity fields contain subsequences that converge to a minimizing pair of the continuous problem in the $H^1$-norm and a mesh-dependent norm, respectively. We present extensive numerical results to illustrate the performance of the approach, including a comparison with the popular Taylor-Hood elements.

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arXiv2025

Artificial Intelligence in Reactor Physics: Current Status and Future Prospects

Reactor physics is the study of neutron properties, focusing on using models to examine the interactions between neutrons and materials in nuclear reactors. Artificial intelligence (AI) has made significant contributions to reactor physics, e.g., in operational simulations, safety design, real-time monitoring, core management and maintenance. This paper presents a comprehensive review of AI approaches in reactor physics, especially considering the category of Machine Learning (ML), with the aim of describing the application scenarios, frontier topics, unsolved challenges and future research directions. From equation solving and state parameter prediction to nuclear industry applications, this paper provides a step-by-step overview of ML methods applied to steady-state, transient and combustion problems. Most literature works achieve industry-demanded models by enhancing the efficiency of deterministic methods or correcting uncertainty methods, which leads to successful applications. However, research on ML methods in reactor physics is somewhat fragmented, and the ability to generalize models needs to be strengthened. Progress is still possible, especially in addressing theoretical challenges and enhancing industrial applications such as building surrogate models and digital twins.

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arXiv2024

Adaptive finite element approximations of the first eigenpair associated with $p$-Laplacian

In this paper, we propose an adaptive finite element method for computing the first eigenpair of the $p$-Laplacian problem. We prove that starting from a fine initial mesh our proposed adaptive algorithm produces a sequence of discrete first eigenvalues that converges to the first eigenvalue of the continuous problem and the distance between discrete eigenfunctions and the normalized eigenfunction set corresponding to the first eigenvalue in $W^{1,p}$-norm also tends to zero. Extensive numerical examples are provided to show the effectiveness and efficiency.

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arXiv2024

Adaptive Computation of Elliptic Eigenvalue Topology Optimization with a Phase-Field Approach

In this paper, we discuss adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two dimensional numerical examples for illustration of efficiency and accuracy. Theoretical findings consist in the vanishing limit of a subsequence of estimators and the convergence of the relevant subsequence of adaptively-generated solutions to a solution to the continuous optimality system.

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arXiv2024

Adversarial neural network methods for topology optimization of eigenvalue problems

This research presents a novel method using an adversarial neural network to solve the eigenvalue topology optimization problems. The study focuses on optimizing the first eigenvalues of second-order elliptic and fourth-order biharmonic operators subject to geometry constraints. These models are usually solved with topology optimization algorithms based on sensitivity analysis, in which it is expensive to repeatedly solve the nonlinear constrained eigenvalue problem with traditional numerical methods such as finite elements or finite differences. In contrast, our method leverages automatic differentiation within the deep learning framework. Furthermore, the adversarial neural networks enable different neural networks to train independently, which improves the training efficiency and achieve satisfactory optimization results. Numerical results are presented to verify effectiveness of the algorithms for maximizing and minimizing the first eigenvalues.

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arXiv2024

Energy stable gradient flow schemes for shape and topology optimization in Navier-Stokes flows

We study topology optimization governed by the incompressible Navier-Stokes flows using a phase field model. Novel stabilized semi-implicit schemes for the gradient flows of Allen-Cahn and Cahn-Hilliard types are proposed for solving the resulting optimal control problem. Unconditional energy stability is shown for the gradient flow schemes in continuous and discrete spaces. Numerical experiments of computational fluid dynamics in 2d and 3d show the effectiveness and robustness of the optimization algorithms proposed.

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arXiv2024

An Adaptive Phase-Field Method for Structural Topology Optimization

In this work, we develop an adaptive algorithm for the efficient numerical solution of the minimum compliance problem in topology optimization. The algorithm employs the phase field approximation and continuous density field. The adaptive procedure is driven by two residual type a posteriori error estimators, one for the state variable and the other for the first-order optimality condition of the objective functional. The adaptive algorithm is provably convergent in the sense that the sequence of numerical approximations generated by the adaptive algorithm contains a subsequence convergent to a solution of the continuous first-order optimality system. We provide several numerical simulations to show the distinct features of the algorithm.

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arXiv2018

Convergence analysis of Galerkin finite element approximations to shape gradients in eigenvalue optimization

Numerical computation of shape gradients from Eulerian derivatives is essential to wildly used gradient type methods in shape optimization. Boundary type Eulerian derivatives are popularly used in literature. The volume type Eulerian derivatives hold more generally, but are rarely noticed and used numerically. We investigate thoroughly the accuracy of Galerkin finite element approximations of the two type shape gradients for optimization of elliptic eigenvalues. Under certain regularity assumptions on domains, we show \emph{a priori} error estimates for the two approximate shape gradients. The convergence analysis shows that the volume integral formula converges faster and generally offers better accuracy. Numerical experiments verify theoretical results for the Dirichlet case. For the Neumann case, however, the boundary formulation surprisingly converges as fast as the volume one. Numerical results are presented.

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