arXiv subjects

Vivien J. Challis

Publications and source records attributed to Vivien J. Challis.

9 recordsLinked to original sources

arXiv2026

Shape calculus and automatic differentiation for multi-phase level-set topology optimisation with unfitted finite elements

We present shape calculus techniques and a scalable automatic shape differentiation framework for multi-phase topology optimisation on unfitted discretisations defined by several level-set functions. First we establish general, exact shape calculus expressions in the discrete case for multi-phase systems by leveraging concepts from convex geometry. To complement this theoretical foundation, we introduce an open-source multi-phase automatic shape differentiation framework based on polytopal cutting. This computational framework is validated against both finite differences and our established exact expressions, matching the latter to near machine precision. Furthermore, the proposed automatic shape differentiation is scalable across distributed computing environments, demonstrating near ideal weak scaling up to 1.65 billion finite elements across 13,824 computer cores. We demonstrate our implementation by solving unfitted multi-phase topology optimisation problems for anisotropic diffusion, linear elasticity, and fluid--structure interaction. Together, these theoretical and computational contributions provide a robust and accessible foundation for advancing multi-phase topology optimisation using unfitted finite element methods. In particular, the methods enable the solution of topology optimisation problems involving multi-phase and multi-physics systems with non-trivial boundary conditions. The open-source software is available at https://github.com/zjwegert/GridapTopOpt.jl.

math.NA↗

arXiv2026

A discontinuous finite element method for the hydroelastic analysis of submerged structures

This paper develops a finite element method for wave-structure interaction problems arising in the hydroelastic modelling of submerged elastic plates. The approach is formulated for both two-dimensional settings and three-dimensional channels. For the fluid part of the problem, we use a symmetric discontinuous Galerkin scheme that captures the discontinuity in the potential across the plate. To impose the Sommerfeld radiation conditions, we take inspiration from the Dirichlet-to-Neumann map approach and use the analytic solution in the semi-infinite domains to construct appropriate non-local boundary conditions. For the plate, we use a continuous/discontinuous Galerkin method to resolve the 4$^{\rm th}$ order operator without using continuously differentiable finite elements. We prove well-posedness of the method, showing that the resulting sesquilinear form is bounded and satisfies the Gårding inequality, leading to a unique and stable solution. We implement our formulation as an open-source tool using the Gridap package in Julia. We validate our method against a hypersingular boundary integral method, finding excellent agreement. The flexibility of the methodology is demonstrated through simulations of a two-dimensional slanted plate and three-dimensional plate geometries including centred rectangular plates, offset rectangular plates, and annulus-shaped plates. The finite element method presented here can be readily applied to model wave energy conversion of piezoelectric bimorphs. Furthermore, the method could easily be extended to consider wave scattering from fixed or rigid structures or from variable bottom topography, and is not limited to hydroelastic applications.

math.NA↗

arXiv2026

Wave energy conversion by floating and submerged piezoelectric bimorph plates

Gaining insight into the interaction between flexible piezoelectric structures and ocean waves can inform the development of compact, high-efficiency wave-energy converters that harvest renewable energy from the marine environment. In this paper, the problem of wave energy absorption by floating and submerged piezoelectric plates is investigated. The equations of motion for a plate consisting of two piezoelectric layers separated by an elastic substrate are derived in dimensional form from the full piezoelectric constitutive laws. A solution using a modal expansion method is proposed, in which the component radiation and diffraction problems are reduced to hypersingular integral equations and solved numerically using a constant panel method. The method is general and can solve the equations of motion for submerged rigid, flexible elastic or flexible piezoelectric plates. Extensive numerical results for the energy absorption and efficiency are given for a range of parameters, including different piezoelectric materials: polyvinylidene fluoride (PVDF) and lead zirconate titanate (PZT-5H). Importantly, greater energy absorption is obtained for submerged plates when compared to plates floating on the surface. Furthermore, clamped boundary conditions give slightly larger energy absorption compared to the simply supported case. Our open-source code is provided at https://github.com/zjwegert/SemiAnalyticWECs.jl.

cond-mat.mtrl-sci↗

arXiv2026

Water wave scattering by a surface-mounted rectangular anisotropic elastic plate

This paper considers the problem of water wave scattering by a rectangular anisotropic elastic plate mounted on the ocean surface, with either free, clamped or simply-supported edges. The problem is obtained as an expansion over the dry modes of the elastic plate, which are computed using a Rayleigh--Ritz method. In turn, the component diffraction and radiation problems are solved by formulating a boundary integral equation and solving numerically using a constant panel method. The results are presented to highlight the resonant responses of the plate under different forcing scenarios. In particular, we illustrate how the excitation of certain modes can be forbidden due to symmetry.

physics.flu-dyn↗

arXiv2025

Level-set topology optimisation with unfitted finite elements and automatic shape differentiation

In this paper we develop automatic shape differentiation techniques for unfitted discretisations and link these to recent advances in shape calculus for unfitted methods. We extend existing analytic shape calculus results to the case where the domain boundary intersects with the boundary of the background domain. We further show that we can recover these analytic derivatives to machine precision regardless of the mesh size using the developed automatic shape differentiation techniques, drastically reducing the burden associated with the analytic derivation of these quantities. In addition, we show that we can also recover the symmetric shape Hessian. We implement these techniques for both serial and distributed computing frameworks in the Julia package GridapTopOpt and the wider Gridap ecosystem. As part of this implementation we propose a novel graph-based approach for isolated volume detection. We demonstrate the applicability of the unfitted automatic shape differentiation framework and our implementation by considering the three-dimensional minimum compliance topology optimisation of a linear elastic wheel and of a linear elastic structure in a fluid-structure interaction problem with Stokes flow. The implementation is general and allows GridapTopOpt to solve a wider range of problems on unstructured meshes without analytic calculation of shape derivatives and avoiding issues that arise when material properties are smoothed at the domain boundary. The software is open source and available at https://github.com/zjwegert/GridapTopOpt.jl.

math.OC↗

arXiv2025

Level set-based inverse homogenisation of three-dimensional piezoelectric materials

In this paper we use memory-distributed level set-based topology optimisation to design three-dimensional periodic piezoelectric materials with enhanced properties. We compare and assess several existing iterative solvers with respect to their weak scalability and find that an approximate Schur complement preconditioned generalized minimal residual method method demonstrates the best performance and scalability for solving the piezoelectric homogenisation equations. We use the developed techniques to computationally design high-resolution piezoelectric metamaterials with enhanced stiffness and piezoelectric properties that yield new insights into material design for sensor, hydrophone, and actuator applications. We suggest two robust structures with no fine-scale features that exhibit enhanced piezoelectric properties several times larger than those of the base material. We find that level set-based topology optimisation is well suited to problems involving piezoelectricity and has the advantage of avoiding large regions of intermediate density material. Our memory-distributed level-set implementation is open source and provided for practitioners in the community.

cs.CE↗

arXiv2024

GridapTopOpt.jl: A scalable Julia toolbox for level set-based topology optimisation

In this paper we present GridapTopOpt, an extendable framework for level set-based topology optimisation that can be readily distributed across a personal computer or high-performance computing cluster. The package is written in Julia and uses the Gridap package ecosystem for parallel finite element assembly from arbitrary weak formulations of partial differential equation (PDEs) along with the scalable solvers from the Portable and Extendable Toolkit for Scientific Computing (PETSc). The resulting user interface is intuitive and easy-to-use, allowing for the implementation of a wide range of topology optimisation problems with a syntax that is near one-to-one with the mathematical notation. Furthermore, we implement automatic differentiation to help mitigate the bottleneck associated with the analytic derivation of sensitivities for complex problems. GridapTopOpt is capable of solving a range of benchmark and research topology optimisation problems with large numbers of degrees of freedom. This educational article demonstrates the usability and versatility of the package by describing the formulation and step-by-step implementation of several distinct topology optimisation problems. The driver scripts for these problems are provided and the package source code is available at https://github$.$com/zjwegert/GridapTopOpt.jl.

cs.MS↗

arXiv2024

Random walk models for the propagation of signalling molecules in one-dimensional spatial networks and their continuum limit

The propagation of signalling molecules within cellular networks is affected by network topology, but also by the spatial arrangement of cells in the networks. Understanding the collective reaction--diffusion behaviour in space of signals propagating through cellular networks is an important consideration for example for regenerative signals that convey positional information. In this work, we consider stochastic and deterministic versions of random walk models of signalling molecules propagating and reacting within one-dimensional spatial networks with arbitrary node placement and connectivity. By taking a continuum limit of the random walk models, we derive an inhomogeneous reaction--diffusion--advection equation, where diffusivity and advective velocity depend on local node density and connectivity within the network. Our results show that large spatial variations of molecule concentrations can be induced by heterogeneous node distributions. Furthermore, we find that noise within the stochastic random walk model is directly influenced by node density. We apply our models to consider signal propagation within the osteocyte network of bone, where signals propagating to the bone surface regulate bone formation and resorption processes. We investigate signal-to-noise ratios for different damage detection scenarios and show that the location of perturbations to the network can be detected by signals received at the network boundaries.

cond-mat.dis-nn↗

arXiv2023

A Hilbertian projection method for constrained level set-based topology optimisation

We present an extension of the projection method proposed by Challis et al. (Int J Solids Struct 45(14$\unicode{x2013}$15):4130$\unicode{x2013}$4146, 2008) for constrained level set-based topology optimisation that harnesses the Hilbertian velocity extension-regularisation framework. Our Hilbertian projection method chooses a normal velocity for the level set function as a linear combination of (1) an orthogonal projection operator applied to the extended optimisation objective shape sensitivity and (2) a weighted sum of orthogonal basis functions for the extended constraint shape sensitivities. This combination aims for the best possible first-order improvement of the optimisation objective in addition to first-order improvement of the constraints. Our formulation utilising basis orthogonalisation naturally handles linearly dependent constraint shape sensitivities. Furthermore, use of the Hilbertian extension-regularisation framework ensures that the resulting normal velocity is extended away from the boundary and enriched with additional regularity. Our approach is generally applicable to any topology optimisation problem to be solved in the level set framework. We consider several benchmark constrained microstructure optimisation problems and demonstrate that our method is effective with little-to-no parameter tuning. We also find that our method performs well when compared to a Hilbertian sequential linear programming method.

math.OC↗