Enforcement Of Boundary Conditions Research Articles

A Finite Operator Learning Technique for Mapping the Elastic Properties of Microstructures to Their Mechanical Deformations

ABSTRACTTo obtain fast solutions for governing physical equations in solid mechanics, we introduce a method that integrates the core ideas of the finite element method with physics‐informed neural networks and concept of neural operators. We propose directly utilizing the available discretized weak form in finite element packages to construct the loss functions algebraically, thereby demonstrating the ability to find solutions even in the presence of sharp discontinuities. Our focus is on micromechanics as an example, where knowledge of deformation and stress fields for a given heterogeneous microstructure is crucial for further design applications. The primary parameter under investigation is the Young's modulus distribution within the heterogeneous solid system. Our investigations reveal that physics‐based training yields higher accuracy compared with purely data‐driven approaches for unseen microstructures. Additionally, we offer two methods to directly improve the process of obtaining high‐resolution solutions, avoiding the need to use basic interpolation techniques. The first one is based on an autoencoder approach to enhance the efficiency for calculation on high resolution grid points. Next, Fourier‐based parametrization is utilized to address complex 2D and 3D problems in micromechanics. The latter idea aims to represent complex microstructures efficiently using Fourier coefficients. The proposed approach draws from finite element and deep energy methods but generalizes and enhances them by learning parametric solutions without relying on external data. Compared with other operator learning frameworks, it leverages finite element domain decomposition in several ways: (1) it uses shape functions to construct derivatives instead of automatic differentiation; (2) it automatically includes node and element connectivity, making the solver flexible for approximating sharp jumps in the solution fields; and (3) it can handle arbitrary complex shapes and directly enforce boundary conditions. We provided some initial comparisons with other well‐known operator learning algorithms, further emphasize the advantages of the newly proposed method.

A boundary enforcement method for the truncated ship structure model during underwater contact explosion experiments

Conducting model experiments is an effective approach for investigating structural damage of ship cabin structures induced by underwater contact explosions. However, large-scale cabin model experiments often encounter challenges in establishing practical constraints, resulting in disparities in boundary constraints between the experimental model and the intact ship model. To overcome this, the present work proposes a novel method for enforcing boundary conditions that involves extending the boundaries of the experimental model, securing the model boundaries with clamps and the rigid wall, and simulating the underwater environment using the temporary water tank. Based on this method, an experiment of a broadside cabin of large ships subjected to underwater contact explosion was conducted, and results are reproduced through the Arbitrary Lagrange-Euler (ALE) method. Subsequently, the explosive load characteristics of the boundary enforcement model are further analyzed, followed by a discussion of the impact of boundary constraints on structural response, thereby validating the effectiveness of the boundary enforcement method.

A Physics-Informed General Convolutional Network for the Computational Modeling of Materials With Damage

Abstract Despite their effectiveness in modeling complex phenomena, the adoption of machine learning (ML) methods in computational mechanics has been hindered by the lack of availability of training datasets, limitations on the accuracy of out-of-sample predictions, and computational cost. This work presents a physics-informed ML approach and network architecture that addresses these challenges in the context of modeling the behavior of materials with damage. The proposed methodology is a novel physics-informed general convolutional network (PIGCN) framework that features (1) the fusion of a dense edge network with a convolutional neural network (CNN) for specifying and enforcing boundary conditions and geometry information, (2) a data augmentation approach for learning more information from a static dataset that significantly reduces the necessary data for training, and (3) the use of a CNN for physics-informed ML applications, which is not as well explored as graph networks in the current literature. The PIGCN framework is demonstrated for a simple two-dimensional, rectangular plate with a hole or elliptical defect in a linear-elastic material, but the approach is extensible to three dimensions and more complex problems. The results presented in this article show that the PIGCN framework improves physics-based loss convergence and predictive capability compared to ML-only (physics-uninformed) architectures. A key outcome of this research is the significant reduction in training data requirements compared to ML-only models, which could reduce a considerable hurdle to using data-driven models in materials engineering where material experimental data are often limited.

Pressure boundary conditions for immersed-boundary methods

Immersed boundary methods have seen an enormous increase in popularity over the past two decades, especially for problems involving complex moving/deforming boundaries. In most cases, the boundary conditions on the immersed body are enforced via forcing functions in the momentum equations, which in the case of fractional step methods may be problematic due to: i) creation of slip-errors resulting from the lack of explicitly enforcing boundary conditions on the (pseudo-)pressure on the immersed body; ii) coupling of the solution in the fluid and solid domains via the Poisson equation. Examples of fractional-step formulations that simultaneously enforce velocity and pressure boundary conditions have also been developed, but in most cases the standard Poisson equation is replaced by a more complex system which requires expensive iterative solvers. In this work we propose a new formulation to enforce appropriate boundary conditions on the pseudo-pressure as part of a fractional-step approach. The overall treatment is inspired by the ghost-fluid method typically utilized in two-phase flows. The main advantage of the algorithm is that a standard Poisson equation is solved, with all the modifications needed to enforce the boundary conditions being incorporated within the right-hand side. As a result, fast solvers based on trigonometric transformations can be utilized. We demonstrate the accuracy and robustness of the formulation for a series of problems with increasing complexity.

A δ-free approach to quantization of transmission lines connected to lumped circuits

The quantization of systems composed of transmission lines connected to lumped circuits poses significant challenges, arising from the interaction between continuous and discrete degrees of freedom. A widely adopted strategy, based on the pioneering work of Yurke and Denker, entails representing the lumped circuit contributions using Lagrangian densities that incorporate Dirac δ-functions. However, this approach introduces complications, as highlighted in the recent literature, including divergent momentum densities, necessitating the use of regularization techniques. In this work, we introduce a δ-free Lagrangian formulation for a transmission line capacitively coupled to a lumped circuit without the need for a discretization of the transmission line or mode expansions. This is achieved by explicitly enforcing boundary conditions at the line ends in the principle of least action. In this framework, the quantization and the derivation of the Heisenberg equations of the network are straightforward. We obtain a reduced model for the lumped circuit in the quantum Langevin form, which is valid for any coupling strength between the line and the lumped circuit. We apply our approach to an analytically solvable network consisting of a semi-infinite transmission line capacitively coupled to an LC circuit and study the behavior of the network as the coupling strength varies.

Weak boundary conditions for Lagrangian shock hydrodynamics: A high-order finite element implementation on curved boundaries

We propose a new Nitsche-type approach for weak enforcement of normal velocity boundary conditions for a Lagrangian discretization of the compressible shock-hydrodynamics equations using high-order finite elements on curved boundaries. Specifically, the variational formulation is appropriately modified to enforce free-slip wall boundary conditions, without perturbing the structure of the function spaces used to represent the solution, with a considerable simplification with respect to traditional approaches. Total energy is conserved and the resulting mass matrices are constant in time. The robustness and accuracy of the proposed method are validated with an extensive set of tests involving nontrivial curved boundaries.

Error analysis of the element-free Galerkin method for a nonlinear plate problem

In this work, we derive a geometrically nonlinear plate model based on the Kirchhoff hypothesis and the large deflection hypothesis, and provide an error analysis of the corresponding element-free Galerkin method. The penalty method is employed to enforce boundary conditions. The error estimate explicitly depends on nodal spacing, number of monomial basis functions, continuity of weight functions, and penalty factors, which provides some practical choices among these key parameters in engineering applications. In addition, we offer guidance on selecting appropriate penalty factors to improve numerical accuracy. Numerical experiments, involving clamped square and circle plates with uniform and nonuniform nodes, as well as a plate model with a corner singularity, are presented to validate the theoretical results.

Adjacency-based, non-intrusive model reduction for vortex-induced vibrations

Vortex-induced Vibrations (VIV) pose computationally expensive problems of high practical interest to several engineering fields. In this work we develop a non-intrusive, reduced-order modeling methodology, applied to two-dimensional (2D), VIV cases subject to a laminar, incompressible flow. Performing a physics-informed approximation of the Arbitrary Lagrangian–Eulerian (ALE) incompressible Navier–Stokes (NS), we derive a discrete-time, quadratic-bilinear model structure for the velocity flowfield on a reference domain. This structure, along with a predefined sparsity pattern motivated by the adjacency-based sparsity of the discretized NS-ALE operators, leads to the formulation of a sparse, full-order model (sFOM) inference problem. Thus, the data-driven inference task requires solving many “local” least squares (LS) problems, isolating the contribution of geometrically “nearest neighbours” for each degree of freedom. Numerical aspects of inference such as data centering and regularization, as well as the direct enforcement of boundary conditions under the sFOM formulation are extensively discussed. The inferred, sFOM operators are then projected to a non-intrusive reduced-order model (ROM) for the velocity flowfield via the Proper Orthogonal Decomposition (POD). The resulting non-intrusive ROM (sFOM-POD) is coupled with the first-principle, 2D solid oscillation equations, resulting to a hybrid physics-informed/first-principle VIV dynamics model, simulated using an implicit time integration scheme. The mapping of the coupled solution from the ALE reference domain to the current configuration is also presented. This methodology is applied to two testcases of an elliptical, non-deformable solid mounted on springs, subject to Re=90,180 flows. Numerical results indicate a successful coupling between the data-driven flowfield and solid dynamics, with prediction errors of less than 3% for both the flowfield and the solid oscillation. A comparative study with respect to the sFOM-POD dimension indicates the robustness and potential of the approach.

Efficient least squares approximation and collocation methods using radial basis functions

We describe an efficient method for the approximation of functions using radial basis functions (RBFs), and extend this to a solver for boundary value problems on irregular domains. The method is based on RBFs with centers on a regular grid defined on a bounding box, with some of the centers outside the computational domain. The equation is discretized using collocation with oversampling, with collocation points inside the domain only, resulting in a rectangular linear system to be solved in a least squares sense. The goal of this paper is the efficient solution of that rectangular system. We show that the least squares problem splits into a regular part, which can be expedited with the FFT, and a low rank perturbation, which is treated separately with a direct solver. The rank of the perturbation is influenced by the irregular shape of the domain and by the weak enforcement of boundary conditions at points along the boundary. The solver extends the AZ algorithm which was previously proposed for function approximation involving frames and other overcomplete sets. The solver has near optimal log-linear complexity for univariate problems, and loses optimality for higher-dimensional problems but remains faster than a direct solver. A Matlab implementation is available at https://gitlab.kuleuven.be/numa/public/2023-RBF-AZalgorithm.

Coupling of smoothed particle hydrodynamics and finite volume method for electrohydrodynamic droplet deformation simulation

In the numerical simulation of droplet electrohydrodynamic deformation, the grid-based method is difficult to handle the large topological deformation of the droplet interface, while the meshless method is challenging in the enforcement of boundary conditions. Therefore, a coupling method of the smoothed particle hydrodynamics (SPH) method and the finite volume method (FVM) is developed in this paper. In the present coupling method, the SPH method is used to solve the governing equations of the two-phase flow, while the FVM is used to solve the governing equations of the electric field. The realization of the coupling method is based on the information transfer between particles and grids. In order to accurately transfer the electric field force of interface grid nodes to SPH particles, a particle splitting interpolation method is developed. The influences of electrical capillary number, electrical conductivity ratio, and permittivity ratio on droplet deformation are studied. The simulation results are in good agreement with the theoretical predictions, and the coupling method can obtain more accurate numerical results than the pure SPH method. The coupling method can provide a reliable way to investigate complex electrohydrodynamic problems.

An improved plate deep energy method for the bending, buckling and free vibration problems of irregular Kirchhoff plates

An improved plate deep energy method without penalty terms is introduced to study the bending, free vibration and buckling problems of irregular Kirchhoff plates. The finite difference produced by convolution operation is employed for a faster calculation of derivatives and enforcement of boundary conditions. Most deep learning based neural network methods adopt the penalty method to impose boundary constraints, which may cause convergence issues especially when dealing with complex solution domain or mixed boundary conditions. In this paper, the R-function is utilized to build the distance function multiplied by the neural network output to meet the Dirichlet boundary constraints, while the boundary constraints expressed by derivative functions are applied by introducing virtual nodes. With such treatments, the boundary losses of a plate system can be avoided. To validate the proposed method, numerical examples considering plates with irregular shapes and mixed boundary conditions are studied, and the results are compared to those provided by the theoretical solutions, finite element method (FEM), and literature.

Volume conservation issue within SPH models for long-time simulations of violent free-surface flows

Smoothed Particle Hydrodynamics (SPH) simulations of violent sloshing flows characterized by a strong fragmentation of free-surface can be affected by volume conservation errors. These errors can accumulate in time and preclude the possibility of using such models for long-time simulations. In the present work, different techniques to measure directly the particles’ volumes by their positions are displayed. Thanks to these algorithms, we discussed how volume errors develop when introducing stabilization terms in the standard SPH schemes. Further, we show that the control of these errors can be achieved through two different (non-alternative) stabilization terms: (i) enforcement of dynamic boundary conditions on thin fluid jets and drops; and (ii) a dynamical switch that can impose or not the time variation of the particle masses.To demonstrate the effectiveness of the numerical techniques, four different violent sloshing flows are investigated and compared with experimental data.

Breakup of a leaky dielectric droplet in a half-sinusoidal wave electric field

In the numerical simulation of droplet breakup under electric field, the meshless method has inherent advantages, however the enforcement of boundary conditions is challenging. Therefore, a coupling method of smoothed particle hydrodynamics(SPH) method and finite volume method(FVM) is proposed to investigate the breakup behavior of leaky dielectric droplet under half-sinusoidal wave electric field. The present numerical method is validated and is well consistent with results in the literature. The effects of electric field strength and droplet viscosity on droplet breakup are mainly discussed, and three different modes of droplet breakup are revealed: conical-conical angle breakup, dumbbell-dimple breakup and cylindrical-non tip breakup. The regions of these three breakup modes are plotted in the breakup phase diagram of electric capillary number and viscosity ratio. The numerical experiments also demonstrate that the droplet breakup mode depends on the velocity field and pressure field of the droplet and the surrounding fluid.

Physics-informed neural networks for mesh deformation with exact boundary enforcement

In this work, we have applied physics-informed neural networks (PINN) for solving mesh deformation problems. We used the collocation PINN method to capture the new positions of the vertex nodes while preserving the connectivity information. We use linear elasticity equations for mesh deformation. To prevent vertex collisions or edge overlap, the mesh movement in this work is conducted in steps with relatively small movements. For moving boundary problems, the exact position of the boundary is essential for having an accurate solution. However, PINNs are frequently unable to satisfy Dirichlet boundary conditions exactly. To overcome this issue, we have used hard boundary condition enforcement to automatically satisfy Dirichlet boundary conditions. Specifically, we first trained a PINN with soft boundary conditions to obtain a particular solution. Then, this solution was tuned with exact boundary positions and a proper distance function by using a new PINN considering only the equation residual. To assess the accuracy of our approach, we used the classical translation and rotation tests and compared them with a proper mesh quality metric considering the change in the element area and shape. The results show the accuracy of this approach is comparable with that of finite element solutions. We also solved different moving boundary problems, resembling commonly used fluid–structure interaction problems. This work provides insight into using PINN for mesh-deformation problems without needing a discretization scheme with reasonable accuracy.

A rotation-free Hellinger-Reissner meshfree thin plate formulation naturally accommodating essential boundary conditions

A rotation-free Hellinger-Reissner meshfree thin plate formulation is proposed to naturally accommodate the essential boundary conditions in a variationally consistent way. In this approach, the Galerkin weak form is established based upon the Hellinger-Reissner variational principle, where the bending moment is expressed as the second order smoothed gradients which inherently embed the integration constraint and fulfill the variational consistency condition. Owing to the Hellinger-Reissner variational principle, the essential boundary conditions naturally arise in the weak form. Accordingly, the enforcement of essential boundary conditions has a similar form as that of the Nitsche's method, i.e., both have standard consistent and stabilized terms. Compared with the Nitsche's method, the costly second and higher order derivatives of traditional meshfree shape functions are replaced by the fast-evaluated second order smoothed gradients and their derivatives. Meanwhile, the stabilized term in proposed method does not involve any artificial parameter and thus eliminates the stabilization parameter-dependent issue in the Nitsche's formulation. Several examples are presented to illustrate the convergence, accuracy and efficiency of the proposed rotation-free Hellinger-Reissner meshfree thin plate formulation.

Modeling nonlinear deformation of slender auxetic structures under follower loads with complex variable meshfree methods

The auxetic structure with negative Poisson’s ratio has broad application prospects in many engineering fields. Although it is often simplified as a linear elastic problem in theoretical models, the generally used slender structure, such as the reentrant honeycombs, will inevitably undergo large deformation resulting in significant non-conservative load effects in the service conditions. Therefore, a numerical framework for modeling nonlinear deformation of the auxetic structures under follower loads with the complex variable element-free Galerkin method is developed in this paper. The application of the complex variable meshfree method is to deal with the numerical difficulties caused by mesh distortion that may occur in large deformation problems, and at the same time, it can improve the construction efficiency of meshfree shape functions through the complex variable moving least-squares approximation. The Galerkin weak form of the incremental total Lagrangian formula for large deformation problems with the enforced essential boundary conditions using the penalty method is derived and then discretized in the complex variable meshless implementation. Five numerical examples are presented with detailed convergence study to demonstrate the accuracy of the proposed approach in dealing with non-conservative large deformation of the slender auxetic structures.

A New Approach to Handle Curved Meshes in the Hybrid High-Order Method

We present here a novel approach to handling curved meshes in polytopal methods within the framework of hybrid high-order methods. The hybrid high-order method is a modern numerical scheme for the approximation of elliptic PDEs. An extension to curved meshes allows for the strong enforcement of boundary conditions on curved domains and for the capture of curved geometries that appear internally in the domain e.g. discontinuities in a diffusion coefficient. The method makes use of non-polynomial functions on the curved faces and does not require any mappings between reference elements/faces. Such an approach does not require the faces to be polynomial and has a strict upper bound on the number of degrees of freedom on a curved face for a given polynomial degree. Moreover, this approach of enriching the space of unknowns on the curved faces with non-polynomial functions should extend naturally to other polytopal methods. We show the method to be stable and consistent on curved meshes and derive optimal error estimates in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} and energy norms. We present numerical examples of the method on a domain with curved boundary and for a diffusion problem such that the diffusion tensor is discontinuous along a curved arc.

Clone particles: A simplified technique to enforce solid boundary conditions in SPH

A simplified technique for the enforcement of boundary conditions along solid profiles is proposed in the framework of the Smoothed Particle Hydrodynamics. The main idea relies on a local definition of the normal and tangent vectors to the solid body, along with a local mirroring of the flow fields. This compound approach allows for a robust and accurate modelling of the fluid–solid interaction close to acute angles and sharp profiles. Numerical benchmarks at increasing complexity are considered to prove the reliability of the proposed technique: hydrostatic test cases, flow past cylinders with elliptic, triangular and spiked circular profile and, finally, a roto-translating motion of a spiked cylinder. The results are compared and validated with other numerical methods.

A Galerkin formulation for the nonlinear analysis of a flexible beam in channel flow

In this paper we propose a Galerkin formulation for the 1-D model describing the nonlinear flow-structure interaction of a flexible beam in confined flow. In broad terms, the system of PDEs is converted into a set of time-dependent equations (ODEs and algebraic) by developing all variables in terms of series of space-dependent orthogonal functions. The beam motion is developed in terms of its mode shapes while the flow pressure and velocity fields in each channel are developed in terms of Chebyshev polynomials. Additionally, a tau-variant of the Galerkin approach enables the enforcement of the nonlinear time-dependent boundary conditions in a well-posed manner. Ultimately, the resulting system is a set of nonlinear differential–algebraic equations that can be truncated at any suitable numbers of terms, leading to exploitable reduced formulations. Compared to CFD methods, this type of formulations is not only more computationally efficient, but also provide an easy discernment of the relevant parameters and often a more intuitive interpretation of results. Convergence studies, in terms of both the truncation of Chebyshev polynomials and beam modes, are performed to access to what extent reduced formulations are viable in different scenarios, including linear stability analysis as well as the calculation of limit-cycle oscillations with or without intermittent impacts between the beam and the side-walls. The presented framework can serve as a basis for a comprehensive analysis of the nonlinear dynamics of flexible beams in confined flow. Namely, it is well adapted to the use of bifurcation analysis tools for the continuation of periodic solutions, which will contribute to a richer understanding of the underlying physics occurring in this type of FSI systems. Moreover, the generic methodology presented here can also be adapted to different systems in the field of fluid–structure interaction, providing compact time-dependent formulations for nonlinear analysis.

A meshfree radial basis function method for simulation of multi‐dimensional conservation problems

AbstractMany computational fluid dynamics problems utilize finite volume frameworks for simulation, due to the simplifications provided by conservative formulation of the driving partial differential equations (PDEs). However, fluid dynamics applications can often involve temporal shifts in the domain structure—such as moving boundaries, or pore structure changes—requiring mesh adaptation throughout computation. These mesh adaptations often render classical numerical methods such as the finite volume method infeasible, due to their reliance on a well‐defined static mesh structure. This limitation has led to the development of a wide variety of meshless methods—techniques that can simulate PDEs without requiring a rigid connective structure between nodes. However, most meshless methods are typically based on finite element or finite difference formulations, and the limited number of meshless finite volume methods (MFVMs) either introduce a weak background mesh, or use weak‐form approximations that do not take full advantage of the strong conservative form of the driving equations. Addressing this gap within this study we outline a meshfree numerical scheme for simulation of partial different equations, based on strong‐form finite volume style formulations. Building upon the previously developed MFVM, this technique uses radial basis functions to interpolate the problem domain, and approximate fluxes in a disjoint finite volume scheme, removing reliance on a mesh structure. We present method derivation, including promising new techniques for enforcing boundary conditions in a meshless environment. Following this we discuss method accuracy and computational performance across a variety of problems in two and three dimensions. We then illustrate how this method may prove beneficial for applications in porous media modeling, and computational fluid dynamics. For completeness, we provide a sensitivity analysis of the method hyper‐parameters and investigate the conservative properties of the method. We also illustrate similarities of this approach to the widely used meshless point collocation methods. We close with a discussion of the strengths, limitations, and broader applicability of the technique.