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Abstract The rapid growth of artificial intelligence is revolutionizing classical engineering society, offering novel approaches to material and structural design and analysis. Among various scientific machine learning techniques, physics-informed neural network (PINN) has been one of the most researched subjects, for its ability to incorporate physics prior knowledge into model training. However, the intrinsic continuity requirement of PINN demands the adoption of domain decomposition when multiple materials with distinct properties exist. This greatly complicates the gradient computation of design features, restricting the application of PINN to structural shape optimization. To address this, we present a novel framework that employs neural network coordinate projection for shape optimization within PINN. This technique allows for direct mapping from a standard shape to its optimal counterpart, optimizing the design objective without the need for traditional transition functions or the definition of intermediate material properties. Our method demonstrates a high degree of adaptability, allowing the incorporation of diverse constraints and objectives directly as training penalties. The proposed approach is tested on magnetostatic problems for iron core shape optimization, a scenario typically plagued by the high permeability contrast between materials. Validation with finite-element analysis confirms the accuracy and efficiency of our approach. The results highlight the framework’s capability as a viable tool for shape optimization in complex material design tasks.

The rapid growth of artificial intelligence is revolutionizing classical engineering society, offering novel approaches to material and structural design and analysis. Among various scientific machine learning techniques, physics-informed neural network (PINN) has been one of the most researched subjects, for its ability to incorporate physics prior knowledge into model training. However, the intrinsic continuity requirement of PINN demands the adoption of domain decomposition when multiple materials with distinct properties exist. This greatly complicates the gradient computation of design features, restricting the application of PINN to structural shape optimization. To address this, we present a novel framework that employs neural network coordinate projection for shape optimization within PINN. This technique allows for direct mapping from a standard shape to its optimal counterpart, optimizing the design objective without the need for traditional transition functions or the definition of intermediate material properties. Our method demonstrates a high degree of adaptability, allowing the incorporation of diverse constraints and objectives directly as training penalties. The proposed approach is tested on magnetostatic problems for iron core shape optimization, a scenario typically plagued by the high permeability contrast between materials. Validation with finite-element analysis confirms the accuracy and efficiency of our approach. The results highlight the framework's capability as a viable tool for shape optimization in complex material design tasks. [ABSTRACT FROM AUTHOR]

A mathematical analysis is performed for a system consisting of two coupled Cahn-Hilliard equations. These equations incorporate a diffusional mobility that depends on concentration. This modeling approach is often used to describe the process of phase separation in a thin layer of a binary liquid mixture covering a substrate, particularly when one of the components wets the substrate. The analysis establishes the existence of a weak formulation for this problem, which is supported by the use of a Lyapunov functional. Additionally, the analysis provides insights into the regularity properties of the weak formulation. [ABSTRACT FROM AUTHOR]

We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qϵ). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qϵ). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qϵ). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces. [ABSTRACT FROM AUTHOR]

Phase field modeling finds utility in various areas. In optimization theory in particular, the distributed control and Neumann boundary control of phase field models have been investigated thoroughly. Dirichlet boundary control in parabolic equations is commonly addressed using the very weak formulation or an approximation by Robin boundary conditions. In this paper, the Dirichlet boundary control for a phase field model with a non-singular potential is investigated using the Dirichlet lift technique. The corresponding weak formulation is analyzed. Energy estimates and problem-specific embedding results are provided, leading to the existence and uniqueness of the solution for the state equation. These results together show that the control to state mapping is well defined and bounded. Based on the preceding findings, the optimization problem is shown to have a solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the speed of moving boundaries for melting micro/nanoparticles in the initial and final stages using asymptotic matchings in a weak formulation of the problem. We find that such a speed is initially proportional to the flux across the moving boundary, however a blowup occurs in a finite time when the surface tension is considered, both numerically and theoretically, by assuming linear relations between thermal conductivities and diffusivities, which paves the way to tackle the related two and higher phase change problems. Last but not least, we verify our theoretical outcomes using a quasi-stationary approximation approach. [ABSTRACT FROM AUTHOR]

In this work, a difference finite element method for the 3D steady Navier--Stokes equations is presented. This new method consists of transmitting the finite element solution (uh, ph) of the three-dimensional (3D) steady Navier--Stokes equations into a series of the finite element solutions (uhnk, phnk) of the 2D steady Oseen iterative equations, which are solved by using the finite element pair (P1b, P1b, P1) × P1 satisfying the discrete inf-sup condition in a 2D domain ω. In addition, we use finite element pair ((P1b, P1b, P1) × (P1 × P0) to solve the 3D steady Oseen iterative equations, where the velocity-pressure pair satisfies the discrete inf-sup condition in a 3D domain Ω under the quasi-uniform mesh condition. To overcome the difficulty of nonlinearity, we apply the Oseen iterative method and present the weak formulation of the difference finite element method for solving the 3D steady τ Oseen iterative equations. Moreover, we provide the existence and uniqueness of the difference finite element solutions (uhn, phn) = (... uhnk∅k(z), ... phnkψk(z)) of the 3D steady Oseen iterative equations and deduce the first order convergence with respect to (σn+1, τ, h) of the difference finite element solutions (uhn, phn) to the exact solution (u, p) of the 3D steady Navier--Stokes equations. Finally, some numerical tests are presented to show the accuracy and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The article proposes an Equivalent Single Layer (ESL) formulation for the linear static analysis of arbitrarily-shaped shell structures subjected to general surface loads and boundary conditions. A parametrization of the physical domain is provided by employing a set of curvilinear principal coordinates. The generalized blending methodology accounts for a distortion of the structure so that disparate geometries can be considered. Each layer of the stacking sequence has an arbitrary orientation and is modelled as a generally anisotropic continuum. In addition, re-entrant auxetic three-dimensional honeycomb cells with soft-core behaviour are considered in the model. The unknown variables are described employing a generalized displacement field and pre-determined through-the-thickness functions assessed in a unified formulation. Then, a weak assessment of the structural problem accounts for shape functions defined with an isogeometric approach starting from the computational grid. A generalized methodology has been proposed to define two-dimensional distributions of static surface loads. In the same way, boundary conditions with three-dimensional features are implemented along the shell edges employing linear springs. The fundamental relations are obtained from the stationary configuration of the total potential energy, and they are numerically tackled by employing the Generalized Differential Quadrature (GDQ) method, accounting for nonuniform computational grids. In the post-processing stage, an equilibrium-based recovery procedure allows the determination of the three-dimensional dispersion of the kinematic and static quantities. Some case studies have been presented, and a successful benchmark of different structural responses has been performed with respect to various refined theories. [ABSTRACT FROM AUTHOR]

• A coupled transient electromagnetic-thermal finite element solution strategy has been developed. • The modeling strategy is suitable to model induction heating for ferromagnetic materials. • The formulation and implementation have been validated against experimental data. Induction heating is used in many industrial applications to heat electrically conductive materials. The coupled electromagnetic-thermal induction heating process is non-linear in general, and for ferromagnetic materials it becomes challenging since both the electromagnetic and the thermal responses are non-linear. As a result of the existing non-linearities, simulating the induction heating process is a challenging task. In the present work, a coupled transient electromagnetic-thermal finite element solution strategy that is appropriate for modeling induction heating of ferromagnetic materials is presented. The solution strategy is based on the isothermal staggered split approach, where the electromagnetic problem is solved for fixed temperature fields and the thermal problem for fixed heat sources obtained from the electromagnetic solution. The modeling strategy and the implementation are validated against induction heating experiments at three heating rates. The computed temperatures, that reach above the Curie temperature, agree very well with the experimental results. [ABSTRACT FROM AUTHOR]

In this article, we propose a mathematical model that describes hydrodynamics and deformation mechanics within a solid tumor which is embedded in or adjacent to a healthy (normal) tissue. The tumor and normal tissue regions are assumed to be deformable and the theory of mixtures is adapted to mass and momentum balance equations for fluid flow and tissue deformation mechanics in each region. The momentum balance equations are coupled via forces that interact between the phases (fluid and solid). Continuity of normal velocities, displacements, and normal stresses along with the Beaver–Joseph–Saffman condition are imposed at the interface between the tumor and tissue regions. The physiological transport parameters (such as hydraulic resistivity or permeability) are assumed to be heterogeneous and deformation dependent which makes the model nonlinear. We establish the existence of a weak solution using Galerkin and weak convergence methods. We show further that the solution is unique and depends continuously on the given data. [ABSTRACT FROM AUTHOR]

Meshless methods have become an essential numerical tool for simulating a wide range of flow–structure interaction problems. However, the way by which the impermeable solid boundary condition is implemented can significantly affect the accuracy of the results and computational cost. This paper develops an improved boundary scheme through a weak formulation for the boundary particles based on Pressure Poisson's Equation (PPE). In this scheme, the wall boundary particles simultaneously satisfy the PPE in the local integration domain by adopting the Meshless Local Petrov–Galerkin method with the Rankine source solution (MLPG_R) integration scheme (Ma, 2005b) and the Neumann boundary condition, i.e., normal pressure gradient condition, on the wall boundary which truncates the local integration domain. The new weak formulation vanishes the derivatives of the unknown pressure at wall particles and is discretized in the truncated support domain without extra artificial treatment. This improved boundary scheme is validated by analytical solutions, numerical benchmarks, and experimental data in the cases of patch tests, lid-driven cavity, flow over a cylinder and monochromic wave generation. Second-order convergent rate is achieved even for disordered particle distributions. The results show higher accuracy in pressure and velocity, especially near the boundary, compared to the existing boundary treatment methods that directly discretize the pressure Neumann boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we derive the weak form for clamped plates composed of incompressible neo-Hookean material from the uniformly valid asymptotic plate theory. By using the finite-element software COMSOL, we study the numerical solutions of the weak form. We show the accuracy and the efficiency of the weak form by comparing the numerical results for the two-dimensional weak form and a three-dimensional model. As a basis for comparison we choose numerical values of the displacement, the second Piola–Kirchhoff stress, and the Green–Lagrange strain at the bottom. The numerical simulations are performed for three different cases of thickness–span ratios, including (1) very thin plate, (2) thin plate, and (3) moderately thick plate. The results show that the uniformly valid plate theory is a reliable and implementable plate theory for even moderately thick plates with large deformations. [ABSTRACT FROM AUTHOR]

In this paper, a difference finite element (DFE) method is presented for the 3D steady Stokes equations. This new method consists of transmitting the finite element solution (u h , p h) of the 3D steady Stokes equations in the direction of (x , y , z) into a series of the finite element solution (u h k , p h k) of the 2D steady Stokes equations. Here the 2D steady Stokes equations are solved by the finite element space pair (P 1 b , P 1 b , P 1) × P 1 , where the 2D finite element pair (P 1 b , P 1 b) × P 1 satisfies the discrete inf-sup condition in a 2D domain ω. Here we design the weak formulation of the DFE method based on the 3D finite element pair ((P 1 b , P 1 b , P 1) × P 1) × (P 1 × P 0) under the quasi-uniform mesh condition, prove that the 3D finite element pair satisfies the discrete inf-sup condition in a 3D domain Ω and provide the existence, uniqueness and stability of the DFE solution (u h , p h) = (∑ k = 0 l 3 u h k ϕ k (z) , ∑ k = 1 l 3 p h k ψ k (z)) and deduce the first order convergence of the DFE solution (u h , p h) with respect to the exact solution (u , p) of the 3D steady Stokes equations. Finally, some numerical tests are presented to show the accuracy and efficiency for the proposed method. [ABSTRACT FROM AUTHOR]

Solving the Helmholtz equation with spatially resolved finite element method (FEM) approaches is a well-established and cost-efficient methodology to numerically predict thermoacoustic instabilities. With the implied zero Mach number assumption all interaction mechanisms between acoustics and the mean flow velocity including the advection of acoustic waves are neglected. Incorporating these mechanisms requires higher-order approaches that come at massively increased computational cost. A tradeoff might be the convective wave equation in frequency domain, which covers the advection of waves and comes at equivalent cost as the Helmholtz equation. However, with this equation only being valid for uniform mean flow velocities it is normally not applicable to combustion processes. The present paper strives for analyzing the introduced errors when applying the convective wave equation to thermoacoustic stability analyses. Therefore, an acoustically consistent, inhomogeneous convective wave equation is derived first. Similar to Lighthill's analogy, terms describing the interaction between acoustics and non-uniform mean flows are considered as sources. For the use with FEM approaches, a complete framework of the equation in weak formulation is provided. This includes suitable impedance boundary conditions and a transfer matrix coupling procedure. In a modal stability analysis of an industrial gas turbine combustion chamber, the homogeneous wave equation in frequency domain is subsequently compared to the Helmholtz equation and the consistent Acoustic Perturbation Equations (APE). The impact of selected source terms on the solution is investigated. Finally, a methodology using the convective wave equation in frequency domain with vanishing source terms in arbitrary mean flow fields is presented. [ABSTRACT FROM AUTHOR]

In this paper, the classical two-dimensional Helmholtz transmission problem is reduced to a local boundary value problem by introducing an artificial boundary. A localized Dirichlet-to-Neumann (DtN) mapping is defined on the artificial boundary. Then the variational equations and Galerkin formulation are derived. The effectiveness of the methods is demonstrated using various numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we find relations between the ionic parameters and the diffusion parameters which are sufficient to ensure the existence of a periodic solution for a well-known monodomain model in a weak sense. We make use of the method of approximation of Faedo-Galerkin to prove the existence of weak periodic solutions of the monodomain model for the electrical activity of the heart assuming that it is periodically activated in its boundaries. Actually, this periodic solution has the same period of activation. Finally, we reflect on how these ionic-diffusive relations are useful to explain the pathophysiology of some rhythm disorders. [ABSTRACT FROM AUTHOR]

The determining problem of the variable characteristics of an elastic rod made of a functionally graded material based on acoustic sounding data is considered. The amplitude-frequency characteristics of the rod during longitudinal and bending vibrations are used as additional information. A system of nonlinear integral equations is derived, the solution of which is based on an iterative scheme. The conditions under which the reconstruction is carried out in a unique way are presented. The results of computational experiments are discussed. [ABSTRACT FROM AUTHOR]

Based on the general linear elasticity relations, an axisymmetric problem on the steady‐state oscillations of a functionally graded hollow cylinder is formulated. The Lamé parameters are considered variable in radial coordinate. Oscillations are caused by the distributed load applied to the outer part of the cylinder boundary. Using the variable separation method, the direct problem on determining the radial and longitudinal components of the displacement field is investigated. The influence of the laws of variation for the Lamé parameters on acoustic characteristics is analysed. The inverse coefficient problem on the identification of the variable Lamé parameters from the data on the amplitude‐frequency characteristic is stated. Based on the weak formulation of the problem for an elastic inhomogeneous body, a general linearised relation for the desired and given characteristics is obtained. A system of the Fredholm integral equations of the first kind is formulated with respect to two unknown corrections to the restored laws of the Lamé parameters change. The solution is built by means of an iterative process. A reconstruction of various laws of changing the Lamé parameters is carried out. The accuracy of the presented algorithm is estimated, and recommendations for the most efficient implementation of the reconstruction procedure are proposed. [ABSTRACT FROM AUTHOR]

• A mixed, low differential order, variational principle for the Foeppl von Karman shell model. • Simple mechanical interpretation of the displacement boundary conditions in terms of the Airy stress function. • FENICS code for its locking-free numerical implementation (Open-acces in a bitbucket repository). A mixed variational principle is proposed for deducing the Föppl–von Kármán equations governing the large deflections of thin elastic plates or shallow shells. Proper boundary conditions are found for the case of applied in-plane tractions and displacements, and simple mechanical interpretations are achieved. Numerical implementation is carried out, along with examples and comparisons with the classical formulation in terms of displacements. [ABSTRACT FROM AUTHOR]

In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair Xh×Mh. The method consists of transmitting the finite element solution (uh,ph) of the 3D steady Navier–Stokes equations into the finite element solution pairs (uhn,phn) based on the finite element space pair Xh×Mh of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair Xh×Mh satisfies the discrete inf-sup condition in a 3D domain Ω. Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution (uhn,phn) of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to (σ,h) of the FE solution (uhn,phn) to the exact solution (u,p) of the 3D steady Navier–Stokes equations in the H1−L2 norm. Finally, we also give the convergence order with respect to (σ,h) of the FE velocity uhn to the exact velocity u of the 3D steady Navier–Stokes equations in the L2 norm.

In this article, we present a biphasic mixture theory based mathematical model for the hydrodynamics of interstitial fluid motion and mechanical behavior of the solid phase inside a solid tumor. The tumor tissue considered here is an isolated deformable biological medium. The solid phase of the tumor is constituted by vasculature, tumor cells, and extracellular matrix, which are wet by a physiological extracellular fluid. Since the tumor is deformable in nature, the mass and momentum equations for both the phases are presented. The momentum equations are coupled due to the interaction (or drag) force term. These governing equations reduce to a one-way coupled system under the assumption of infinitesimal deformation of the solid phase. The well-posedness of this model is shown in the weak sense by using the inf-sup (Babuska–Brezzi) condition and Lax–Milgram theorem in 2D and 3D. Further, we discuss a one-dimensional spherical symmetry model and present some results on the stress fields and energy of the system based on 𝐿2 and Sobolev norms. We discuss the so-called phenomena of “necrosis” inside a solid tumor using the energy of the system.

This article considers a plane strain gradient plasticity theory of the Gurtin–Anand model [M. Gurtin, L. Anand, A theory of strain gradient plasticity for isotropic, plastically irrotational materials Part I: Small deformations, J. Mech. Phys. Solids 53 (2005), 1624–1649] for an isotropic material undergoing small deformation in the absence of plastic spin. It is assumed that the system of microstresses is purely dissipative, so that the free energy reduces to a function of the elastic strain, while the microstresses are only related to the plastic strain rate and gradient of the plastic strain rate via the constitutive relations. The plane strain problem of the Gurtin–Anand model for a purely dissipative process gives rise to elastic incompressibility. A weak formulation of the flow rule is derived, making the plane strain problem suitable for finite element implementation.

Numerical solutions to partial differential equations (PDEs) are instrumental for material structural design where extensive data screening is needed. However, traditional numerical methods demand significant computational resources, highlighting the need for innovative optimization algorithms to streamline design exploration. Direct gradient-based optimization algorithms, while effective, rely on design initialization and require complex, problem-specific sensitivity derivations. The advent of machine learning offers a promising alternative to handling large parameter spaces. To further mitigate data dependency, researchers have developed physics-informed neural networks (PINNs) to learn directly from PDEs. However, the intrinsic continuity requirement of PINNs restricts their application in structural mechanics problems, especially for composite materials. Our work addresses this discontinuity issue by substituting the PDE residual with a weak formulation in the physics-informed training process. The proposed approach is exemplified in modeling digital materials, which are mathematical representations of complex composites that possess extreme structural discontinuity. This article also introduces an interactive process that integrates physics-informed loss with design objectives, eliminating the need for pretrained surrogate models or analytical sensitivity derivations. The results demonstrate that our approach can preserve the physical accuracy in data-free material surrogate modeling but also accelerates the direct optimization process without model pretraining., (© The Author(s) 2024. Published by Oxford University Press on behalf of National Academy of Sciences.)

We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing–hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging duality: It is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry‐Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing–hedging duality for American options. [ABSTRACT FROM AUTHOR]

Finite Element Limit Analysis (FELA) is increasingly used for calculating the ultimate bearing capacity of structures made of ductile materials. Within FELA for reinforced concrete structures the elements have been based on rigorous lower bound or boundary mixed formulations. The lower bound element formulation may be overly constrained for certain meshes and the dual displacement interpretation may contain spurious modes, moreover the boundary mixed element formulation has an internal equilibrium node with no associated displacement field. Here a consistent and general weak formulation based on virtual work is presented specifically for both the lower and the upper bound problem, and it is shown that they are each others dual ensuring uniqueness of the optimal solution. As a consequence there is no difference between the solutions based on the weak upper and lower bound methods. Here a plane element is presented, with a linear stress variation and a quadratic displacement field, optionally including a concentrated bar element with a linear variation of the normal force. These elements are applied in a verification example and two reinforced concrete examples where they show very good results for both load level, stress distribution and collapse mechanism even for coarse meshes. • Weak formulations for lower and upper bound methods showing duality and uniqueness. • Weak formulations for second order cone programming to be solved by convex solvers. • An accurate triangular element with reinforced concrete yield criteria is formulated. • Numerical examples show accurate limit loads compared to existing methods. [ABSTRACT FROM AUTHOR]

Finite difference procedures for flux determination are not well suited for application to field results obtained from finite element calculations. A novel method for flux calculation has been derived. This method is based on the weak formulation and is suitable for use with finite elements. A matrix formulation for local and global application to finite element formulations is presented. An additional benefit of the method is that Neumann boundary conditions can be easily incorporated in the finite element formulation of the nonlinear field problem. A comparison between the finite difference, Pade derivative and novel finite element procedures is demonstrated through one- and two-dimensional examples.

We consider a relaxation of the viscous Cahn-Hilliard equation induced by the second-order inertial term utt. The equation also contains a semilinear term f(u) of “singular” type. Namely, the function f is defined only on a bounded interval of R corresponding to the physically admissible values of the unknown u, and diverges as u approaches the extrema of that interval. In view of its interaction with the inertial term utt, the term f(u) is diffcult to be treated mathematically. Based on an approach originally devised for the strongly damped wave equation, we propose a suitable concept of weak solution based on duality methods and prove an existence result.