Petrov-Galerkin Approach Research Articles

Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics

While deep learning and data-driven modeling approaches based on deep neural networks (DNNs) have recently attracted increasing attention for solving partial differential equations, their practical applications to real-world scientific and engineering problems remain limited due to the relatively low accuracy and high computational costs. In this study, we present a differentiable programming-based hybrid meshfree approach within the field of computational mechanics. This approach seamlessly integrates neural network-based deep learning architectures with physics-based meshfree discretization techniques, thus referred to as the neural-integrated meshfree (NIM) method. To effectively approximate the solution, NIM employs a hybrid scheme called NeuroPU approximation, which combines continuous DNN representations with partition of unity (PU) basis functions associated with the underlying spatial discretization. This neural-numerical hybridization not only enhances the solution representation through functional space decomposition but also reduces both the size of DNN model and the need of automatic differentiation for spatial gradient computations, leading to a significant improvement in training efficiency. Under the NIM framework, we propose two truly meshfree solvers: the strong form-based NIM (S-NIM) and the local variational form-based NIM (V-NIM). In the S-NIM solver, the strong-form governing equation is directly considered in the loss function, while the V-NIM solver employs a local Petrov–Galerkin approach that allows the construction of variational residuals based on arbitrary overlapping subdomains. This ensures both the satisfaction of underlying physics and the preservation of the meshfree property. We perform extensive numerical experiments on both stationary and transient benchmark problems to assess the effectiveness of the proposed NIM methods in terms of accuracy, scalability, generalizability, and convergence properties. Moreover, comparative analysis with other physics-informed machine learning methods demonstrates that NIM, especially V-NIM, significantly enhances both accuracy and efficiency in end-to-end predictive capabilities.

A cell-based smoothed finite element model for the analysis of turbulent flow using realizable k-ε model and mixed meshes

Smoothed finite element method (S-FEM) has been widely employed in computational mechanics, particularly in computational solid mechanics (CSM) and, to a lesser extent, in computational fluid dynamics (CFD). The present work focuses on the development of S-FEM for solving three-dimensional incompressible turbulent flow problems using the realizable k-ε model. The Streamline Upwind Petrov-Galerkin approach combined with Stabilized Pressure Gradient Projection (SUPG/SPGP) was used to restrain/control the spatial oscillation and instability in conjunction with mixed meshes employed to discretize complex geometries. The presented turbulent cell-based S-FEM (CS-FEM) was validated under several common flow types, including mesh turbulence, turbulent channel flow, turbulent flow past a circular cylinder, flow separation around an obstacle, flow in the dimpled channel, and flow in the upper human airway. In particular, the presented work exhibited the latent capability of the CS-FEM in solving turbulent flows near boundary regions and strong convection regions. Meanwhile, the results of the CS-FEM were compared to those of the lowest equal-order finite element method (i.e. P1–P1 or Q1–Q1 elements) in simulating turbulent flows, and it was found that the CS-FEM had better agreement with other published works.

Application of Meshless local Petrov-Galerkin approach for steady state groundwater flow modeling

Abstract The complicated behavior of groundwater system in an arid aquifer is generally studied by solving the governing equations using either analytical or numerical methods. In this regard, analytical methods are just for some aquifers with regular boundaries. Numerical methods used for this aim are finite difference (FDM) and finite element methods (FEM) which are engaged for some simple aquifers. Using them in the complex cases with irregular boundaries has some shortcomings, dependent on meshes. In this study, meshless local Petrov-Galerkin (MLPG) method based on the moving kriging (MK) approximation function is used to simulate groundwater flow in steady state over three aquifers, two standard and a real field aquifer. Moving kriging function known as new function which reduces the uncertain parameter. For the first aquifer, a simple rectangular aquifer, MLPG-MK indicates good agreement with analytical solutions. In the second one, aquifer conditions get more complicated. However, MLPG-MK reveals results more accurate than FDM. RMSE for MLPG-MK and FDM is 0.066 and 0.322 m respectively. In the third aquifer, Birjand unconfined aquifer located in Iran is investigated. In this aquifer, there are 190 extraction wells. The geometry of the aquifer is irregular as well. With this challenging issues, MLPG-MK again shows satisfactory accuracy. As the RMSE for MLPG-MK and FDM are 0.483 m and 0.566 m. therefore, planning for this aquifer based on the MLPG-MK is closer to reality.

Mesh distortion insensitive and locking‐free Petrov–Galerkin low‐order EAS elements for linear elasticity

AbstractOne of the most successful mixed finite element methods in solid mechanics is the enhanced assumed strain (EAS) method developed by Simo and Rifai in 1990. However, one major drawback of EAS elements is the highly mesh dependent accuracy. In fact, it can be shown that not only EAS elements, but every finite element with a symmetric stiffness matrix must either fail the patch test or be sensitive to mesh distortion in bending problems (higher order displacement modes) if the shape of the element is arbitrary. This theorem was established by MacNeal in 1992. In the present work we propose a novel Petrov–Galerkin approach for the EAS method, which is equivalent to the standard EAS method in case of regular meshes. However, in case of distorted meshes, it allows to overcome the mesh‐distortion sensitivity without loosing other advantages of the EAS method. Three design conditions established in this work facilitate the construction of the element which does not only fulfill the patch test but is also exact in many bending problems regardless of mesh distortion and has an exceptionally high coarse mesh accuracy. Consequently, high quality demands on mesh topology might be relaxed.

On the Energy Stable Approximation of Hamiltonian and Gradient Systems

Abstract A general framework for the numerical approximation of evolution problems is presented that allows to preserve an underlying dissipative Hamiltonian or gradient structure exactly. The approach relies on rewriting the evolution problem in a particular form that complies with the underlying geometric structure. The Galerkin approximation of a corresponding variational formulation in space then automatically preserves this structure which allows to deduce important properties for appropriate discretization schemes including projection based model order reduction. We further show that the underlying structure is preserved also under time discretization by a Petrov–Galerkin approach. The presented framework is rather general and allows the numerical approximation of a wide range of applications, including nonlinear partial differential equations and port-Hamiltonian systems. Some examples will be discussed for illustration of our theoretical results, and connections to other discretization approaches will be highlighted.

The Meshfree Localized Petrov-Galerkin Approach in Slope Stability Analysis

Abstract The article focuses on the use of the meshfree numerical method in the field of slope stability computations. There are many meshfree implementations of numerical methods. The article shows the results obtained using the meshfree localized Petrov-Galerkin method (MLPG) – localized weak-form of the equilibrium equations with an often used elastoplastic material model based on Mohr-Coulomb (MC) yield criterion. The most important aspect of MLPG is that the discretization process uses a set of nodes instead of elements. Node position within the computational domain is not restricted by any prescribed relationship. The shape functions are constructed using just the set of nodes present in the simple shaped domain of influence. The benchmark slope stability numerical model was performed using the developed meshfree computer code and compared with conventional finite element (FEM) and limit equilibrium (LEM) codes. The results showed the ability of the implemented theoretical preliminaries to solve the geotechnical stability problems.

Active tuning and maximization of natural frequency in three-dimensional functionally graded shape memory alloy composite structures using meshless local Petrov–Galerkin method

A mesh-less method based on the local Petrov–Galerkin approach is developed and numerically implemented for active tuning and three-dimensional (3D) analysis of free vibration response of functionally graded (FG) shape memory alloy (SMA) sandwich composites. The local symmetric weak formulation is derived based on the 3D equilibrium equations of elasticity and the field variables are interpolated using the moving-least squares approach. Two main goals are followed. Firstly, free vibration analysis of a sandwich plate made of a SMA fiber reinforced composite core and two FG face sheets are presented. A parametric study is performed to investigate the influences of design parameters on the fundamental natural frequency (FNF) of composite structure and also maximize that. Secondly, SMA wires are embedded to fiber reinforced (FR) composite structures to enable precisely active control of their dynamic characteristics and change that in a required manner. To increase the controllability level and tune the complex exhibits, functionally graded shape memory alloys are considered which have the advantage of combining the functionalities of the shape memory effect and those of FG structures. This paper uses multi-layer activation patterns and the polynomial graded orientation angle of SMA wire along the structure thickness to furnish the FR composite designer with a powerful free vibration control tool. Finally, two activation techniques of SMA wires, the active property tuning (APT) and the active strain energy tuning (ASET) techniques, are compared. It is concluded from all investigated cases that the obtained FNF of ASET technique is always larger than the APT technique.

The Petrov–Galerkin finite element method for the numerical solution of time-fractional Sharma–Tasso–Olver equation

In this paper, time-fractional Sharma–Tasso–Olver (STO) equation has been solved numerically through the Petrov–Galerkin approach utilizing a quintic B-spline function as the test function and a linear hat function as the trial function. The Petrov–Galerkin technique is effectively implemented to the fractional STO equation for acquiring the approximate solution numerically. The numerical outcomes are observed in adequate compatibility with those obtained from variational iteration method (VIM) and exact solutions. For fractional order, the numerical outcomes of fractional Sharma–Tasso–Olver equations are also compared with those obtained by variational iteration method (VIM) in Song et al. [Song L., Wang Q., Zhang H., Rational approximation solution of the fractional Sharma–Tasso–Olver equation, J. Comput. Appl. Math. 224:210–218, 2009]. Numerical experiments exhibit the accuracy and efficiency of the approach in order to solve nonlinear fractional STO equation.

A meshless local Petrov–Galerkin method for solving the neutron diffusion equation

The goal of this study is to solve the neutron diffusion equation by using a meshless method and evaluate its performance compared to traditional methods. This paper proposes a novel method based on coupling the meshless local Petrov–Galerkin approach and the moving least squares approximation. This computational procedure consists of two main steps. The first involved applying the moving least squares approximation to construct the shape function based on the problem domain. Then, the obtained shape function was used in the meshless local Petrov–Galerkin method to solve the neutron diffusion equation. Because the meshless method is based on eliminating the mesh-based topologies, the problem domain was represented by a set of arbitrarily distributed nodes. There is no need to use meshes or elements for field variable interpolation. The process of node generation is simply and fully automated, which can save time. As this method is a local weak form, it does not require any background integration cells and all integrations are performed locally over small quadrature domains. To evaluate the proposed method, several problems were considered. The results were compared with those obtained from the analytical solution and a Galerkin finite element method. In addition, the proposed method was used to solve neutronic calculations in the small modular reactor. The results were compared with those of the citation code and reference values. The accuracy and precision of the proposed method were acceptable. Additionally, adding the number of nodes and selecting an appropriate weight function improved the performance of the meshless local Petrov–Galerkin method. Therefore, the proposed method represents an accurate and alternative method for calculating core neutronic parameters.

On the solution of time-fractional KdV–Burgers equation using Petrov–Galerkin method for propagation of long wave in shallow water

In the present article, Petrov–Galerkin method has been utilized for the numerical solution of nonlinear time-fractional KdV–Burgers (KdVB) equation. The nonlinear KdV–Burgers equation has been solved numerically through the Petrov–Galerkin approach utilising a quintic B-spline function as the trial function and a linear hat function as the test function . The numerical outcomes are observed in good agreement with exact solutions for classical order. In case of fractional order, the numerical results of KdV–Burgers equations are compared with those obtained by new method proposed in [1]. Numerical experiments exhibit the accuracy and efficiency of the approach in order to solve nonlinear dispersive and dissipative problems like the time-fractional KdV–Burgers equation.

A meshless local Petrov–Galerkin approach for solving the convection-dominated problems based on the streamline upwind idea and the variational multiscale concept

ABSTRACTA meshless local Petrov–Galerkin (MLPG) approach based on the streamline upwind (SU) idea and the variational multiscale (VMS) concept, called as VMS-SUMLPG method, is herein proposed to solve the convection-dominated problems. In the present VMS-SUMLPG method, the streamline upwind is constructed in the test function to solve the non-self-adjoint matrix. Meanwhile, the VMS concept as a stability term is adopted to alleviate the numerical instability such as spurious oscillations, overshoots, and undershoots. Its numerical accuracy and stability are validated by comparing with the streamline upwind Petrov–Galerkin (SUMLPG) method and the finite volume method with high-order difference schemes for two classical convection-dominated problems at the Peclet number ranging from 106 to 108. It is shown that the numerical solutions of the present VMS-SUMLPG method are accuracy, smoothness, and stability.

Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation

In this article, modi ed Korteweg-de Vries (mKdV) equation is solved numerically by using lumped Petrov-Galerkin approach, where weight functions are quadratic and the element shape functions are cubic B-splines. The proposed numerical scheme is tested by applying four test problems including single solitary wave, interaction of two and three solitary waves, and evolution of solitons with the Gaussian initial condition.In order to show the performance of the algorithm, the error norms, L2, L1, and a couple of conserved quantities are computed. For the linear stability analysis of numerical algorithm, Fourier method is also investigated. As a result, the computed results show that the presented numerical scheme is a successful numerical technique for solving the mKdV equation. Therefore, the presented method is preferable to some recent numerical methods.

A theoretical study of COmpRessed SolvING for advection-diffusion-reaction problems

We present a theoretical analysis of the CORSING (COmpRessed SolvING) method for the numerical approximation of partial differential equations based on compressed sensing. In particular, we show that the best s-term approximation of the weak solution of a PDE with respect to a system of N trial functions, can be recovered via a Petrov-Galerkin approach using m << N test functions. This recovery is guaranteed if the local a-coherence associated with the bilinear form and the selected trial and test bases fulfills suitable decay properties. The fundamental tool of this analysis is the restricted inf-sup property, i.e., a combination of the classical inf-sup condition and the well-known restricted isometry property of compressed sensing.

Modeling of porous piezoelectric structures by the meshless local Petrov-Galerkin method

ABSTRACTA meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary value problems of porous piezoelectric solids. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors in both solid and fluid phases. Stationary and transient 2-D and 3-D axisymmetric problems are considered in this article. Nodal points are spread on the problem domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares scheme. After performing the spatial integration, one obtains a system of ordinary differential equations for certain nodal unknowns. The resulting system is solved numerically by the Houbolt finite-difference scheme as a time stepping method. The proposed method is applied to bending problems associated with a porous piezoelectric 2-D plate and 3-D axisymmetric cylinder under simply supported and clamped boundary conditions.

Finite-element modeling of thermal aspects in high speed cold strip rolling

An integrated model based on finite-element method has been proposed to examine the mechanical and thermal responses of strips and work-rolls in tandem and reverse cold rolling operations. The model has been developed such that the influence of various process parameters, such as lubrication, rolling speed, frictional state and back-up rolls, can be examined. Thermal behaviors of the rolled material and the work-rolls have been analyzed using stream-line upwind Petrov–Galerkin approach, in order to make the model applicable to high-speed rolling processes, as well. The results have been compared to the actual on-line measurements and shown to be of acceptable accuracy. Such modeling approach can be considered as a useful means, providing a detailed insight on the thermo-mechanical response of strips, as well as work-rolls, during high-speed cold rolling of steel strips. Additionally, special attention has been drawn towards the prediction of the occurrence of the metallurgical phenomenon dynamic strain aging based on the results obtained from the numerical modeling in this work.

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Abstract Solving the Helmholtz equation for a large number of input data in an heterogeneous media and unbounded domain still represents a challenge. This is due to the particular nature of the Helmholtz operator and the sensibility of the solution to small variations of the data. Here a reduced order model is used to determine the scattered solution everywhere in the domain for any incoming wave direction and frequency. Moreover, this is applied to a real engineering problem: water agitation inside real harbors for low to mid-high frequencies. The proper generalized decomposition (PGD) model reduction approach is used to obtain a separable representation of the solution at any point and for any incoming wave direction and frequency. Here, its applicability to such a problem is discussed and demonstrated. More precisely, the contributions of the paper include the PGD implementation into a perfectly matched layer framework to model the unbounded domain, and the separability of the operator which is addressed here using an efficient higher-order projection scheme. Then, the performance of the PGD in this framework is discussed and improved using the higher-order projection and a Petrov–Galerkin approach to construct the separated basis. Moreover, the efficiency of the higher-order projection scheme is demonstrated and compared with the higher-order singular value decomposition.

Analyses of Circular Magnetoelectroelastic Plates with Functionally Graded Material Properties

A meshless method based on the local Petrov–Galerkin approach is proposed for plate bending analysis with material containing functionally graded magnetoelectroelastic properties. Material properties are considered to be continuously varying along the plate thickness. Axial symmetry of geometry and boundary conditions for a circular plate reduces the original 3D boundary value problem into a 2D problem in axial cross section. Both stationary and transient dynamic conditions for a pure mechanical load are considered in this article. The local weak formulation is employed on circular subdomains in the axial cross section. Subdomains surrounding nodes are randomly spread over the analyzed domain. The test functions are taken as unit step functions in derivation of the local integral equations (LIEs). The moving least-squares (MLS) method is adopted for the approximation of the physical quantities in the LIEs. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time-stepping method.

Numerical Modelling of Melt Behaviour in the Lower Vessel Head of a Nuclear Reactor

This paper describes progress on a consistent approach for multi-phase flow modelling with phase-change. Although, the de- veloped methods are general purpose the applications presented here cover the LIVE experiments involving core melt phenomena at the lower vessel head of a nuclear reactor. These include corium pool formation, coolability and solidification. A new method for solving the compositional multi-phase flow equations to calculate material indicator fields is adopted. An interface-capturing scheme based on high-order accurate compressive advection methods including a Petrov-Galerkin approach is employed to main- tain sharpness of the interfaces between materials. A novel control volume-finite element mixed formulation based on the P1DG-P2 (linear discontinuous in velocity and quadratic continuous in pressure) element pair which can uniquely ensure that key balances are maintained in the equations is developed to discretise the governing equations in space. Anisotropic mesh adaptivity is used to focus the numerical resolution around the interfaces and other areas of important dynamics.

Mindlin theory for the bending of porous plates

Biot’s poroelastic theory has been applied for Mindlin plates to model moderately thick plates. If Mindlin’s kinematical assumptions and a power series expansion for the pore pressure in the thickness direction are considered, the original 3D problem is reduced to 2D. A truncated power series expansion with quadratic terms is considered for the pore pressure. Two functional relationships based on the given boundary conditions and one PDE are derived for the expansion coefficients. A meshless method based on the local Petrov–Galerkin approach is proposed to solve the set of governing PDE in the neutral plane of the poroelastic plate. All in-plane quantities are approximated by the moving least-squares scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns.

Bending of a porous piezoelectric cylinder under a thermal load

A meshless method based on the local Petrov–Galerkin approach is proposed to analyze bending of a porous piezoelectric cylinder under thermal loading. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors for solid and fluid phases. The influence of thermal expansion coefficients in solid and fluid phases on the plate deflection and on the induced electric potential is investigated via the local integral equation method developed in this paper. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares (MLS) scheme. The heat conduction equation is considered as uncoupled with respect to the mechanical and electrical fields.