Imposition Of Boundary Conditions Research Articles

Neural PDE Solvers for Irregular Domains

Neural network-based approaches for solving partial differential equations (PDEs) have recently received special attention. However, most neural PDE solvers only apply to rectilinear domains and do not systematically address the imposition of boundary conditions over irregular domain boundaries. In this paper, we present a neural framework to solve partial differential equations over domains with irregularly shaped (non-rectilinear) geometric boundaries. Given the shape of the domain as an input (represented as a binary mask), our network is able to predict the solution field, and can generalize to novel (unseen) irregular domains; the key technical ingredient to realizing this model is a physics-informed loss function that directly incorporates the interior-exterior information of the geometry. We also perform a careful error analysis which reveals theoretical insights into several sources of error incurred in the model-building process. Finally, we showcase various applications in 2D and 3D, along with favorable comparisons with ground truth solutions.

Embedded finite volume technique for fluid/rigid-body interaction problems

This work describes a formulation for the treatment of submerged rigid solids within the framework of the finite volume method. The proposal consists in considering the solid as a porous region and the hydrodynamic forces that arise from the Darcy terms. The methodology is implemented in an open code that allows large-scale calculations. In the present work, we present the methodology and its verification using cases of study, to demonstrate spatio-temporal numerical convergence and the accuracy of the imposition of boundary conditions. For this purpose, cases that have an analytical solution are analyzed, such as the Pouiseuille flow, the first and second Stokes problems, and immersed walls that move with imposed movement generating different pressures between both faces. A moving cylinder analysis is shown to validate the numerical technique in comparison with cases reported in the literature. The analysis leads to stability conditions against external parameters such as permeability. The ability of the method to represent moving walls and immersed bodies with imposed movement is established.

Lagrange multiplier imposition of non-conforming essential boundary conditions in implicit material point method

The Material Point Method (MPM) is an established and powerful numerical method particularly useful for simulating large-scale, rapid soil deformations. Therefore, it is often used for the numerical investigation of mass movement hazards such as landslides, debris flows, or avalanches. It combines the benefits of both mesh-free and mesh-based continuum-based discretization techniques by discretizing the physical domain with Lagrangian moving particles carrying the history-dependent variables while the governing equations are solved at the Eulerian background grid, which brings many similarities with commonly used finite element methods. However, due to this hybrid nature, the material boundaries do not usually coincide with the nodes of the computational grid, which complicates the imposition of boundary conditions. Furthermore, the position of the boundary may change at each time step and, moreover, may be defined at arbitrary locations within the computational grid that do not necessarily coincide with the body contour, leading to different interactions between the material and the boundary. To cope with these challenges, this paper presents a novel element-wise formulation to weakly impose non-conforming Dirichlet conditions using Lagrange multipliers. The proposed formulation introduces a constant Lagrange multiplier approximation within the constrained elements in combination with a methodology to eliminate superfluous constraints. Therefore, in combination with simple element-wise interpolation functions classically utilized in MPM (and FEM) to approximate the unknown field, a suitable Lagrange multiplier discretization is obtained. In this way, we obtain a robust, efficient, and user-friendly boundary imposition method for immersed methods specified herein for implicit MPM. Furthermore, the extension to frictionless slip conditions is derived. The proposed methodologies are assessed by comparing the numerical results with both analytical and experimental data to demonstrate their accuracy and wide range of applications.

Finite element interpolated neural networks for solving forward and inverse problems

We propose a general framework for solving forward and inverse problems constrained by partial differential equations, where we interpolate neural networks onto finite element spaces to represent the (partial) unknowns. The framework overcomes the challenges related to the imposition of boundary conditions, the choice of collocation points in physics-informed neural networks, and the integration of variational physics-informed neural networks. A numerical experiment set confirms the framework’s capability of handling various forward and inverse problems. In particular, the trained neural network generalises well for smooth problems, beating finite element solutions by some orders of magnitude. We finally propose an effective one-loop solver with an initial data fitting step (to obtain a cheap initialisation) to solve inverse problems.

Energy ratio response at the interface of elastic and dual-porous thermoelastic half-spaces

The present study delves into the intricate examination of energy distribution inherent in plane waves interfacing with an elastic half-space and a thermoelastic half-space characterized by a dual porosity framework. Employing the memory-dependent dual-phase-lag (DPL) hyperbolic two-temperature (H2T) thermoelastic paradigm, the investigation encompasses various incident wave types. The governing equations, rendered in a non-dimensional format, are meticulously addressed by applying the rigorous technique of eigenmode analysis. The intricate energy ratios are meticulously ascertained through the judicious imposition of boundary conditions and the discerning employment of reflection and transmission coefficients. Graphical representations have been exhibited, elucidating the effects of diverse parameters on distinct energy ratios within crystalline structures akin to magnesium materials. These parameters encompass but are not confined to the H2T paradigm, the absence of the two-temperature influence, the classical two-temperature approach, memory effects, and a spectrum of distinct kernel functions. The proposed model emanates cross-disciplinary utility, traversing the domains of seismology, acoustics, optics, materials science, structural engineering, and geophysics.

PhySR: Physics-informed deep super-resolution for spatiotemporal data

High-fidelity simulation of complex physical systems is exorbitantly expensive and inaccessible across spatiotemporal scales. Recently, there has been an increasing interest in leveraging deep learning to augment scientific data based on the coarse-grained simulations, which is of cheap computational expense and retains satisfactory solution accuracy. However, the major existing work focuses on data-driven approaches which rely on rich training datasets and lack sufficient physical constraints. To this end, we propose a novel and efficient spatiotemporal super-resolution framework via physics-informed learning (i.e., PhySR), inspired by the independence between temporal and spatial derivatives in partial differential equations (PDEs). The general principle is to leverage the temporal interpolation for flow estimation, and then introduce convolutional-recurrent neural networks for learning temporal refinement. Furthermore, we employ the stacked residual blocks with wide activation and sub-pixel layers with pixelshuffle for spatial reconstruction, where feature extraction is conducted in a low-resolution latent space. Moreover, we consider hard imposition of boundary conditions in the network to improve reconstruction accuracy. Results demonstrate the superior effectiveness and efficiency of the proposed method, which outperforms existing baseline algorithms, based on extensive numerical experiments.

Insights on using the boundary integral SPH formulations to calculate Laplacians with Dirichlet boundaries

Handling boundary conditions in Smooth Particle Hydrodynamics (SPH) has been known as a challenging problem for a long time, especially when particle approximation is implemented for truncated kernels on the boundary. This work develops a novel particle consistent gradient formulation to impose the Dirichlet boundary condition on a Laplacian with boundary integrals. Boundary integral formulation allows the accurate imposition of boundary conditions without creating virtual particles to extend boundaries, as used in conventional SPH modeling practices. Here, boundary integral formulations are presented for Laplacian operators and the imposition of Dirichlet boundary conditions on arbitrary boundary shapes. We also explore using this particle consistent first-order derivative twice to evaluate the entire Laplacian. Further, the convergence characteristics of different Laplacian formulations with boundary integrals are discussed. Finally, numerical experiments solving explicit thermal and fluid problems are presented to demonstrate the robustness of various Laplacian operators for practical use.

Non‐iterative Green's function approach for 2D unbounded heterogeneous fluid‐saturated media

AbstractThe Green's function approach (GFA) developed in previous work uses the classical Green's function for point load (Kelvin's fundamental solution, Melan's fundamental solution, and others) and the reciprocity theorem to capture changes in the displacement, strain, and stress fields of a geological formation subjected to the processes of extraction or injection of fluids. The great advantage of this method compared to the classical finite element method (FEM) is that it does not require the imposition of boundary conditions and the problem analysis can be performed considering only the reservoir or other regions of interest. In the original version of the GFA, the displacement field is calculated using an iterative numerical scheme, which decreases the computational performance of the method and may present convergence problems. Such limitations have made it difficult to use the GFA in real problems. The present work proposes a non‐iterative numerical scheme capable of expanding the applicability of GFA and, simultaneously, improving its computational performance. The results presented in this work demonstrate that using this numerical scheme, the GFA consumes up to 17.5 times less CPU time compared to iterative scheme and this relationship can be even greater if the heterogeneity of the material increases. Using a geological profile constructed from seismic images of the Brazilian pre‐salt (Tupi field located in the Santos basin), it is shown that the non‐iterative GFA allows the analysis of complex geological formations.

Weak imposition of boundary conditions for the boundary element method in the time domain

In this article we present a boundary element formulation for time dependent elastodynamic wave propagation in 2D interior domains with inhomogeneous boundary conditions. These are imposed in a weak sense, allowing the unified implementation of a wide range of boundary data, from pure Dirichlet to pure Neumann and mixed ones. Numerical experiments indicate that the considered approach is effective and reliable. We discuss several numerical results which, compared to those obtained by the strong imposition of the boundary data, highlight the equivalence in the behavior of the error.

Immersed boundary based near-wall modeling for large eddy simulation of turbulent wall-bounded flow

This paper describes the coupling of the immersed boundary method and the near-wall modeling of large eddy simulation for high Reynolds number turbulent flows over complex geometries on Cartesian grids. To overcome the spurious oscillation problem arising from the imposition of boundary conditions on the stair-case off-wall boundaries, several key ingredients have been employed, such as interpolating the friction velocity instead of the flow velocity from near-wall fluid interior points, evaluating the gradients by the weighted least square method and correcting the wall-normal gradient of the tangential velocity with the law of the wall at the off-wall boundaries. Furthermore, a hybrid RANS-LES approach has been applied to the near-wall eddy viscosity, either through an empirical blending function or the Reynolds stress balance constraint. We systematically discuss the effects of the hybrid eddy viscosity in a turbulent channel flow and a high lift three-element airfoil. Enforcing the Reynolds stress balance constraint turns out to be very robust in the considered cases. For the high lift three-element airfoil flow, the overall wall pressure and skin friction are predicted reasonably well and smoothly. The flow details are in a good agreement with the experimental data as well.

An Improved Neural Particle Method for Complex Free Surface Flow Simulation Using Physics-Informed Neural Networks

The research on free surface flow is of great interest in fluid mechanics, with the primary task being the tracking and description of the motion of free surfaces. The development of numerical simulation techniques has led to the application of new methods in the study of free surface flow problems. One such method is the Neural Particle Method (NPM), a meshless approach for solving incompressible free surface flow. This method is built on a Physics-Informed Neural Network (PINN), which allows for training and solving based solely on initial and boundary conditions. Although the NPM is effective in dealing with free surface flow problems, it faces challenges in simulating more complex scenarios due to the lack of additional surface recognition algorithms. In this paper, we propose an improved Neural Particle Method (INPM) to better simulate complex free surface flow. Our approach involves incorporating alpha-shape technology to track and recognize the fluid boundary, with boundary conditions updated constantly during operation. We demonstrate the effectiveness of our proposed method through three numerical examples with different boundary conditions. The result shows that: (1) the addition of a surface recognition module allows for the accurate tracking and recognition of the fluid boundary, enabling more precise imposition of boundary conditions in complex situations; (2) INPM can accurately identify the surface and calculate even when particles are unevenly distributed. Compared with traditional meshless methods, INPM offers a better solution for dealing with complex free surface flow problems that involve random particle distribution. Our proposed method can improve the accuracy and stability of numerical simulations for free surface flow problems.

A Meshfree Approach Based on Moving Kriging Interpolation for Numerical Solution of Coupled Reaction-Diffusion Problems

In this paper, a meshfree approach based on moving kriging interpolation is presented for numerical solution of coupled reaction-diffusion problems. The proposed approach is developed based upon local collocation using moving Kriging shape function. It is truly meshless and having the Kronecker delta property for accurate imposition of boundary conditions. In the proposed model, the weight function is used with correlation parameter treated as the model internal length factor. This produces a local moving kriging method with improved accuracy together with an ease to choose the weight function factor. The method can hence be used in an efficient manner without cumbersome effort for choosing its parameter. The meshless approach is presented for the first time for numerical solution of reaction-diffusion systems. Problems of Turing system and pattern formation in several 2D domains are solved in this study. The efficacy and accuracy of the proposed method for the reaction-diffusion systems in different problem domains are presented in comparison to available exact solution and other numerical methods. It is found that the present method is accurate and effective as a computational procedure for solving reaction-diffusion problems.

A novel local Hermite radial basis function‐based differential quadrature method for solving two‐dimensional variable‐order time fractional advection–diffusion equation with Neumann boundary conditions

AbstractA novel Hermite radial basis function‐based differential quadrature (H‐RBF‐DQ) method is presented in this paper based on 2D variable order time fractional advection–diffusion equations with Neumann boundary conditions. The proposed method is designed to treat accurately for derivative boundary conditions, which considerably improve the approximation results and extend the range of applicability for the method of RBF‐DQ. The advantage of the present method is that the Hermite interpolation coefficients are only dependent of the point distribution yielding a substantially better imposition of boundary conditions, even for time evolution. The proposed algorithm is thoroughly validated and is demonstrated to handle the fractional calculus problems with both Dirichlet and Neumann boundaries very well.

The imposition of nonconforming Neumann boundary condition in the material point method without boundary representation

Since its original formulation, the material point method (MPM) has been repeatedly further developed to become a powerful tool for simulating large deformation problems. However, the imposition of boundary condition in the MPM remains a challenge since most material points are not located at the boundary and regular background mesh is most commonly used. Therefore, the existing methods to impose boundary condition always require a boundary tracking or reconstruction algorithm, which is not always efficient or accurate. In the current work, a novel strategy to apply nonconforming Neumann boundary condition is proposed that does not require the exact boundary position. The original problem with a nonconforming traction boundary condition is transformed into an equivalent problem with a prescribed virtual stress field. Importantly, both the original and the transformed problems produce the exact same response within the material domain. To solve the equivalent problem, a modified governing equation without boundary representation is subsequently constructed, where the volume integral terms are computed by both particle quadrature and cell quadrature. Several numerical examples are investigated to assess the accuracy and demonstrate the capability of the proposed approach in simulating different engineering problems. These numerical examples include mesh refinement to illustrate the method’s good convergence and a 3D complex surface to illustrate the method’s capability.

Local maximum-entropy approximation based stabilization methods for the convection diffusion problems

Local Maximum Entropy (LME) approximation in the field of elliptic partial differential equations is more advanced compared to convection dominated flow problems. This paper presents a comparative study on the LME based meshfree stabilization methods for convection dominated convection-diffusion problems. LME approximants account for the disposal of the elements and direct imposition of boundary conditions, whereas the stabilization methods are meant to deal with the issues associated with the non-self-adjoint convective term. A new meshfree stabilization scheme, residual free bubbles based multiscale element free Galerkin method is also presented and compared with streamline upwind Petrov Galerkin meshfree method and variational multiscale element free Galerkin Method. The effect of different priors and radius of support which define the LME basis are studied to test their importance in the three methods with respect to standard benchmark problems. Cubic spline prior based LME is found to be enough to produce better stability and accuracy for all the stabilization methods when compared to quartic spline and Gaussian priors. LME based variational multiscale element free Galerkin method is observed to yield better stability and accuracy for all the priors. The convergence rate for the three stabilization methods is observed to be in good agreement with the classical FEM-SUPG.

A deforming‐mesh finite‐element approach applied to the large‐translation and free‐surface scenario of fused deposition modeling

AbstractA numerical study of the fused deposition modeling (FDM) process using a boundary‐conforming free‐surface finite element approach is performed. Due to the complexity of the FDM process, among all of its parts, we focus on the deposition and spreading of an individual filament. The polymer behavior, that is, the shear rate dependent and temperature‐dependent viscosity, is included by the Cross‐WLF viscosity model. The moving domain is addressed by the virtual region mesh update method, which, in the present article, is extended to free‐surface problems. The particularity of dividing the mesh domain into an activated and a deactivated domain makes it possible to handle large translatory mesh deformation. In this work, we make use of the level of detail offered by a boundary‐conforming approach regarding both topology accuracy and the imposition of boundary conditions in order to study the deposition of a single filament at a small scale. Parameters with a direct impact on the mechanical properties of the final object can be straightforwardly computed by a boundary‐conforming approach, for instance, the cross‐section, the contact area, the temperature distribution, and the heat fluxes over the surfaces. The presented approach is validated by a two‐dimensional benchmark test case before the numerical results of the three‐dimensional simulation of the filament deposition are shown.

A meshless thermal modelling for functionally graded porous materials under the influence of temperature dependent heat sources

In this study, a meshless thermal modeling with meshless local moving kriging interpolation method is presented for analysis of functionally graded porous (FGP) materials under the influence of temperature dependent heat sources. The method is developed based on local collocation with moving kriging shape function. It is truly meshless and having the Kronecker delta property for accurate imposition of boundary conditions. In the proposed model, the weight function is used with correlation parameter treated as the model internal length factor. This produces a local moving kriging method with improved accuracy together with an ease to choose the weight function factor. The method can hence be used in an efficient manner without cumbersome effort for choosing its parameter. The contribution is highlighted in the present study, which is still unexplored in detail in literature. Different forms of porosity are considered to investigate the effect of porosity to heat transfer performance of FGP materials subjected to temperature dependent heat sources. In addition, the method is applied for thermal analysis of FGP materials considering arbitrary geometries, mixed boundary conditions and varied power-law index. Numerical results show the effectiveness and accuracy of the meshless method for thermal modeling and analysis of FGP materials.

Computational framework for resolving boundary layers in electrochemical systems using weak imposition of Dirichlet boundary conditions

We present a finite element based computational framework to model electrochemical systems. The electrochemical system is represented by the coupled Poisson–Nernst–Planck (PNP) and Navier–Stokes (NS) equations. The key quantity of interest in such simulations is the current (flux) at the system boundaries. Accurately computing the current flux is challenging due to the small critical dimension of the boundary layers (small Debye layer) that require fine mesh resolution at the boundaries. We present a numerical framework which resolves this challenge by utilizing a weak imposition of Dirichlet boundary conditions for Poisson–Nernst–Planck equations. In this numerical framework we utilize a block iterative strategy to solve NS and PNP equations. This allows us to efficiently and easily implement the weak imposition of Dirichlet boundary conditions. The results from our numerical framework shows excellent agreement when compared to strong imposition of boundary conditions (strong imposition requires a much finer mesh). Furthermore, we show that the weak imposition of the boundary conditions allows us to resolve the fluxes in the boundary layers with much coarser meshes compared to strong imposition. We also show that the method converges as we refine the mesh near the boundaries at a much faster rate compared to strong imposition of the boundary layer. We present multiple test cases with varying boundary layer thickness to illustrate the utility of the numerical framework. We illustrate the approach on canonical 3D problems that otherwise would have been computationally intractable to solve accurately. Lastly, we simulate electrokinetic instabilities near a perm-selective membrane with weakly imposed boundary conditions on the membrane. This approach substantially reduces the computational cost of modeling thin boundary layers in electrochemical systems.

Weak Versus Strong Wall Boundary Conditions for the Incompressible Navier-Stokes Equations

The pressure-velocity formulation of the incompressible Navier-Stokes equations is solved using high-order finite difference operators satisfying a summation-by-parts property. Two methods for imposing Dirichlet boundary conditions (one strong and one weak) are presented and proven stable using the energy method. Additionally, novel diagonal-norm second-derivative finite difference operators are derived with highly improved boundary accuracy. Accuracy and convergence measurements are presented and verified against theoretical expectations. Numerical experiments also show that subtle effects close to solid walls are more efficiently captured with strong boundary condition imposition methods rather than weak (less degrees of freedom required).

Meshless thermal modeling and analysis of functionally graded materials using a meshless local moving kriging interpolation method

In this study, meshless thermal analysis of functionally graded materials (FGM) is presented by using a meshless local moving kriging interpolation method. The method is developed based on local collocation with moving kriging shape function. It is truly meshless and having the Kronecker delta property for accurate imposition of boundary conditions. To improve its accuracy, the weight function is used with correlation parameter treated as the model internal length factor. This produces a meshless local method with clear meaning of model parameter, improved accuracy and allowing a more straightforward analysis for parametric study. Effects of weight function type, parameter value and number of supporting nodes to the method accuracy and convergence rate are first examined in a detailed parametric study of a benchmark problem. Based on the study, suitable parameter, weight function and effective range for supporting nodes number are elucidated. The proposed method is then applied for complex thermal analysis of FGM in arbitrary geometries, taking into account for both temperature and spatial dependent material properties and varied power-law index, including the presence of porosity as well. Numerical results show the effectiveness and accuracy of the meshless method for thermal modeling and analysis of FGM.