Petrov Galerkin Research Articles

Error analysis of Petrov-Galerkin immersed finite element methods

This paper designs and analyzes a new and stable Petrov–Galerkin (PG) immersed finite element method (IFEM) for the second-order elliptic interface problems by introducing stabilization terms based on the classical PG-IFEM, which lacks the local positivity. The Petrov–Galerkin immersed finite element method uses the immersed finite element functions for the trial space and the standard finite element functions for the test space. Both the a priori and a posteriori error estimates of the method are analyzed in this paper. We prove the continuity and inf-sup condition and the a priori error estimate of the energy norm. The proposed a posteriori error estimator is proved to be both reliable and efficient, with both reliability and efficiency constants independent of the location of the interface. Extensive numerical results confirm the numerical scheme’s optimal convergence and indicate the robustness with respect to the interface-mesh intersection and the coefficient contrast, despite the robustness of the inf-sup constant with respect to the interface-mesh intersection has yet been theoretically proved.

Hourglassing‐ and locking‐free mesh distortion insensitive Petrov–Galerkin EAS element for large deformation solid mechanics

AbstractWe present a novel geometrically nonlinear EAS element with several desirable features. First, a Petrov–Galerkin ansatz significantly improves the element's performance in distorted meshes without loosing the simple strain‐driven format. Second, the recently proposed mixed integration point strategy is employed to improve the element's robustness in the Newton–Raphson scheme. Finally and most importantly, we enhance the spatial displacement gradient instead of the usually modified deformation gradient. This allows to construct an element without the well‐known spurious instabilities in compression and tension as shown numerically and supported by a corresponding hypothesis. All in all, this leads to a robust, stable, locking‐free, and mesh distortion insensitive finite element successfully applied in a wide range of examples.

Reprint of: Robust and globally efficient reduction of parametric, highly nonlinear computational models and real time online performance

This paper considers the problem of parametric, nonlinear, projection-based model order reduction (PMOR) and makes three contributions to advance its state of the art. First, it presents a computational strategy for reducing the total offline cost of nonlinear PMOR by enabling hyperreduction to be performed adaptively within the typical greedy sampling procedure, rather than uniformly at each of its iteration. Then, it presents a more accurate and computationally efficient approach for training hyperreduction based on residual Jacobians rather than residuals. Finally, it demonstrates for a large-scale, industrial-grade, parametric, nonlinear application, that the two aforementioned contributions make both offline and online stages of nonlinear PMOR practical in today’s engineering environments. The paper also shows that despite the so-called Kolmogorov n-width barrier for the model reduction of convection-dominated transport problems, the current state of the art of nonlinear, Petrov–Galerkin PMOR equipped with the advances proposed in this paper is capable of accelerating by orders of magnitude the accurate solution of shape-parametric, high Reynolds number, viscous flow problems represented by the Reynolds-averaged Navier–Stokes equations augmented with turbulence modeling. All three contributions are presented and discussed in the contexts of steady-state CFD problems; however, they are equally applicable to unsteady CFD problems as well as problems in solid mechanics and structural dynamics.

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.

Stabilized and variationally consistent integrated meshfree formulation for advection-dominated problems

Meshfree methods such as reproducing kernel (RK) approximation are suitable for modeling fluid flow problems because of their flexibility in controlling local smoothness and order of basis, as well as straightforward construction of higher-order gradients. However, Eulerian-described partial differential equations often suffer from numerical instability in the solution of Bubnov–Galerkin​ methods because of the non-self-adjoint advection terms. This is true for both mesh-based and meshfree methods. Although stabilized Petrov–Galerkin formulation, such as the variational multiscale method, has addressed this issue, a careless selection of the numerical quadrature can still result in variational inconsistency in the Galerkin weak form, which leads to a suboptimal convergence. Strong advection could also amplify the unstable modes from a reduced quadrature. This study provides a variationally consistent (VC) approach to correct the loss of Galerkin exactness in nodally integrated meshfree modeling for the advection diffusion equation. A gradient stabilization method is proposed to enhance the coercivity of the system. Several numerical examples are provided to verify the effectiveness and efficiency of the proposed approaches in modeling advection dominated problems.

Simulation of reactive transport in porous media using Meshless Local Petrov Galerkin (MLPG) and combination of Meshless Weak and Strong (MWS) form methods

In this paper, two meshless methods, namely, a weak form Meshless Local Petrov Galerkin (MLPG) method, and Meshless Weak Strong (MWS) form method, obtained by combining MLPG with a strong form Radial Point Collocation Method (RPCM), are presented for simulation of advection-dispersion-reaction phenomena of the contaminants in the porous media. The first-order decay and sorption reactions are considered in this study. The Crank Nicolson scheme is applied for the time discretization. The weak form MLPG is a truly meshless and robust numerical technique, that can be applied to complex aquifer systems with derivative boundaries. However, in this method, the computational time is increased due to the integration, which is not essential for simple problems. Thus, the MLPG method is further coupled with a strong form RPCM with an aim of decreasing the background integration, by modelling only the nodes around the derivative boundaries using MLPG method and the other nodes by a direct RPCM which do not require integration. The proposed MWS model automatically converts into a complete RPCM model if there are no derivative boundaries. Thus, this model being both accurate and computationally efficient is suitable for simple and moderately complex aquifer systems and MLPG is the most stable and reliable method for modelling the most complex aquifer problems. Both the developed models are tested with available analytical solutions and applied for hypothetical case studies. The results prove the efficiency of the models and the applicability of each model is described in detail.

A strong-form meshfree computational method for plane elastostatic equations of anisotropic functionally graded materials via multiple-scale Pascal polynomials

A strong-form meshfree method is proposed for solving plane elastostatic equations of anisotropic functionally graded materials. Any general function may be the grading function and it is changing smoothly from location to location in the material. The proposed method is based on Pascal polynomial basis and multiple-scale technique and it is a genuinely meshfree method since no numerical integrations over domains and meshing processes are required for considered problems. Implementation of the proposed method is straightforward and the method gives very accurate results. Stability of the solutions are examined numerically in occurrence of random noise. Some certain test problems with known exact solutions are solved both on regular and irregular geometries. Acquired solutions by the suggested method are compared with the exact solutions as well as with solutions of some existing numerical techniques in literature, such as boundary element, meshless local Petrov–Galerkin and radial basis function based meshless methods, to show accuracy of the proposed method.

Uniform stability of the SUPG method for the evolutionary convection–diffusion–reaction equation

To solve the evolutionary convection-diffusion-reaction equation, the implicit Euler method and the streamline upwind Petrov Galerkin (SUPG) method are used for time discretization and space discretization respectively in this paper. Uniform stability is obtained with respect to the temporal and spatial discretization parameters. A new projection is introduced for the stability analysis instead of the common idea of material derivative. The analysis does not require a strict relation between time step and space step, which is usually needed by derivations based on the technique of material derivative. Furthermore, it can handle the case of time-dependent convection and reaction coefficients.

How to lighten parametric inverse computations if the misfit is non-convex and the forward solver needs stabilization

Solving parametric inverse problems (IPs) for partial differential equations (PDEs) plays an important role in medical diagnosis, resource investigation, nondestructive testing and many other human activities. Real-world IPs formulated as misfit functional minimization tasks are frequently ill-conditioned. The origins of this ill-posedness are the misfit multimodality and insensitivity as well as the instability of the utilized numerical PDE solver. We propose a complex multi-population memetic strategy HMS combined with the Petrov–Galerkin method stabilized by the Demkowicz operator to overcome these obstacles. The paper delivers a rigorous mathematical formulation of the strategy as well as a theoretical motivation for common inverse/forward error scaling, which significantly reduces the computational cost of the strategy. The presented theory is illustrated with two examples. The first one shows the analytical construction of the Demkowicz operator. The second one is an application of HMS in solving an IP utilizing the approximate stabilization with a forward solver that uses the isogeometric residual minimization (iGRM) method.

On the performance of a Chimera-FEM implementation to treat moving heat sources and moving boundaries in time-dependent problems

Problems with moving sources and moving inner boundaries in transient regime are of high interest in many research fields and engineering applications. One approach to properly tackle such problems is based on the Chimera method for non-matching grids, where each moving object is defined on a fine mesh that moves across the fixed coarse background. In this way, the high gradients around moving sources or boundaries are captured by the fine mesh without need of globally fine fixed meshes or adaptive refinement. In this work, Chimera is implemented in the framework of the finite element method, following an arbitrary Lagrangian–Eulerian formulation for the moving meshes and a standard Eulerian formulation for the fixed background mesh. Further, in case of convection-dominated problems, the scheme is stabilized using the streamline upwind Petrov–Galerkin method. The coupling between the moving fine meshes and the coarse fixed one is achieved via Dirichlet boundary conditions and a high-order interpolation algorithm. The performance of the proposed methodology in terms of accuracy and stability is assessed by means of numerical tests to be compared with equivalent problems solved using fixed meshes. Further tests serve to highlight the good performance of the proposed Chimera-based finite element method to address both convection-dominated problems with multiple moving boundaries and sources, and three-dimensional arc welding processes. • An algebraic-based Chimera-FEM scheme for unsteady problems is proposed. • Remeshing is avoided for large movements of heat sources and boundaries. • Advective-dominant problems can be addressed with Chimera-FEM and SUPG. • Good accuracy is obtained compared to conventional one-domain approaches. • The scheme is suitable to be used on 3D welding process simulations.

Totally volume integral of fluxes for discontinuous Galerkin method-two dimensional Euler equations

In this paper, the scheme of constructing high-order accurate totally volume discontinuous finite element method for the numerical solution of the 1D Euler equations is extended to 2D Euler equations on Cartesian meshes. In the present work, the boundary integral fluxes are transformed into volume integral by applying divergence theorem to the boundary integral of the Riemann fluxes. Therefore, the totally volume discontinuous finite element is independent on the boundary integral fluxes at the element boundaries as opposed to the classical discontinuous Galerkin method. The accuracy is obtained by applying high-order polynomial approximations within elements using the tensor product of Lagrange polynomial. For temporal integration, strong stability preserving Runge-Kutta method SSPRK (3, 3) is applied. The scheme is stabilized by using Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. For the spatial discretization, the polynomial of order 1and 2 are used, the shape function is constructed for the master (computational) element after applying the coordinate transformation for the physical domain, the transformation for the governing equations is performed to get it in the function of computational Cartesian, then the governing equations are put in conservative form. The numerical results of applying totally volume integral discontinuous Galerkin method for two-dimensional Euler equations presented in this paper show that the scheme is very accurate, fast, and effective even with shock appearance.

Three-dimensional coupling between Boussinesq (FEM) and Navier–Stokes (particle based) models for wave structure interaction

The paper presents coupling between a mesh-based finite-element model for Boussinesq equations (FEBOUSS Agarwal et al., 2022) with a meshless local Petrov–Galerkin model for the Navier–Stokes equations (MLPG_R Agarwal et al., 2021) in 3D. Boussinesq equation models are widely used for simulating wave-propagation over large domains with uneven topography using a 2D surface mesh. Mesh-less models inherently capture large free-surface deformations and have shown promise in simulating wave-structure interaction, run-up and breaking phenomenon. The hybrid approach in this paper assumes a 3D MLPG_R sub-domain surrounded by the 2D mesh of FEBOUSS. The coupling interface in MLPG_R consists of relaxation zones that can be placed along multiple boundaries of the sub-domain for exchanging particle velocity from FEBOUSS. This hybrid model is therefore capable of simulating directional waves, that has not been reported previously.The paper first presents the procedure for calculating the depth-resolved velocities in 3D from the Boussinesq model. The resultant velocities are compared against theory, experiments and other models. The following sections present the coupling algorithm along a single and multiple coupling interfaces in MLPG_R. Validation results for this hybrid model are provided using surface elevation and velocity measurements for regular waves, including directional cases. In general, the results from the hybrid model are reported to have marginal over-prediction of peaks compared to purely MLPG_R simulation. Finally, the interaction of a vertical cylinder with direction regular wave is simulated using the 3D hybrid model.

Error analysis of a SUPG-stabilized POD-ROM method for convection-diffusion-reaction equations

A reduced order model (ROM) method based on proper orthogonal decomposition (POD) is analyzed for convection-diffusion-reaction equations. The streamline-upwind Petrov–Galerkin (SUPG) stabilization is used in the practically interesting case of dominant convection, both for the full order method (FOM) and the ROM simulations. The asymptotic choice of the stabilization parameter for the SUPG-ROM is done as proposed in the literature. This paper presents a finite element convergence analysis of the SUPG-ROM method for errors in different norms. The constants in the error bounds are uniform with respect to small diffusion coefficients. An approach for a posteriori error estimation is discussed. Numerical studies illustrate the performance of the SUPG-ROM method.

A Boundary Integral Formulation and a Topological Energy-Based Method for an Inverse 3D Multiple Scattering Problem with Sound-Soft, Sound-Hard, Penetrable, and Absorbing Objects

Abstract In this paper, we study numerical methods for simulating acoustic scattering by multiple three-dimensional objects of different nature (penetrable, sound-soft, sound-hard and absorbing targets) simultaneously present in the background media. We derive and analyze a boundary integral system of equations that arises when the solution of the problem is represented via single-layer potentials. We give abstract necessary and sufficient conditions for convergence of Petrov–Galerkin discretizations and show that spectral methods satisfy these conditions. Superalgebraic convergence order of the discrete method for smooth objects is illustrated in some test cases. After that, we tackle the inverse problem of finding the shape of objects of different unknown nature from measurements of the total field at a set of receptors. We propose a numerical algorithm based on the computation of the topological energy of a weighted multifrequency least squares cost functional and present some numerical examples to illustrate its capabilities.

Shape optimization with immersed interface finite element method

AbstractShape Optimization using conventional interface‐fitted finite element method requires either mesh morphing or remeshing, which is computationally expensive and prone to errors due to suboptimal mesh or solution interpolation. In this article, we present a shape optimization scheme which uses a four‐noded rectangular immersed‐interface element circumventing the need for interface‐fitted mesh for finite element analysis, and mesh morphing or re‐meshing for shape optimization. The analysis problem is solved using a non‐conformal, Petrov–Galerkin (ncPG) formulation. We use non‐conformal trial functions with conformal test functions. A fixed structured grid with a linear approximation for the interface within each element is used. We perform a bi‐material patch test to confirm the consistency and convergence of the immersed‐interface finite element method (IIFEM). The convergence of the displacement and stress error norms are of the same or slightly better order compared to the interface‐fitted FEM. Thus, IIFEM reduces the cost of meshing without compromising the accuracy of the solutions obtained. We then perform analytical design sensitivity analysis to obtain the gradient of the global stiffness matrix with respect to shape design variables. The sensitivities are used in the gradient‐based optimization formulation of the shape design problem. We present several ways to parametrize the design space. Finally, verification and case studies are presented to demonstrate the accuracy and potential of the approach.

Extending FEniCS to work in higher dimensions using tensor product finite elements

We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is structured as a Cartesian product of two lower dimensional subdomains. In this study we consider Dirichlet problems for scalar linear partial differential equations, though the methodology can be extended to non-linear problems. The utilization of tensor product finite elements allows us to construct a global system of linear algebraic equations that only relies on the finite element infrastructure of the lower dimensional subdomains contained in FEniCS. We demonstrate the effectiveness of our methodology in four distinctive test cases. The first test case is a Poisson equation posed in a four dimensional domain which is a Cartesian product of two unit squares solved using the classical Galerkin finite element method. The second test case is the wave equation in space–time, where the computational domain is a Cartesian product of a two dimensional space grid and a one dimensional time interval. In this second case we also employ the Galerkin method. The third test case is an advection dominated advection–diffusion equation where the global domain is a Cartesian product of two one dimensional intervals in which the streamline upwind Petrov–Galerkin method is applied to ensure discrete stability. The final test case uses the Galerkin approach to solve a Poisson problem on a Cartesian product of two intervals with a spatially varying, non-separable diffusivity term In all cases, a p=1 basis is used and optimal L2 convergence rates of order hp+1 of the errors are achieved with respect to h refinement.

Combining DPG in space with DPG time-marching scheme for the transient advection–reaction equation

In this article, we present a general methodology to combine the Discontinuous Petrov–Galerkin (DPG) method in space and time in the context of methods of lines for transient advection–reaction problems. We first introduce a semidiscretization in space with a DPG method redefining the ideas of optimal testing and practicality of the method in this context. Then, we apply the recently developed DPG-based time-marching scheme, which is of exponential-type, to the resulting system of Ordinary Differential Equations (ODEs). We also discuss how to efficiently compute the action of the exponential of the matrix coming from the space semidiscretization without assembling the full matrix. Finally, we verify the proposed method for 1D+time advection–reaction problems showing optimal convergence rates for smooth solutions and more stable results for linear conservation laws comparing to the classical exponential integrators.

A DPG method for shallow shells

We develop and analyze a discontinuous Petrov–Galerkin method with optimal test functions (DPG method) for a shallow shell model of Koiter type. It is based on a uniformly stable ultraweak formulation and thus converges robustly quasi-uniformly. Numerical experiments for various cases, including the Scordelis–Lo cylindrical roof, elliptic and hyperbolic geometries, illustrate its performance. The built-in DPG error estimator gives rise to adaptive mesh refinements that are capable to resolve boundary and interior layers. The membrane locking is dealt with by raising the polynomial degree only of the tangential displacement trace variable.

Two-sided dimension reduction for linear control systems of [formula omitted]th-order form

In this article, a two-sided structure-preserving dimension reduction method for linear control systems of kth-order form is discussed. We first recall the Petrov–Galerkin framework of dimension reduction for the kth linear systems. Then the moments of systems at a given frequency point s0 are discussed. According to the recurrence form of the moments, we obtain the input Krylov subspace and the projection matrix V. Via dually designing output Krylov subspace, we obtain the output Krylov subspace basis matrix W. Hence, the two-sided dimension reduction method is obtained via combination V with W. Afterwards one can obtain the result that the number of moments matched is twice as much as the column dimension of the projection matrix. Finally, one numerical experiment is performed to verify the proposed dimension reduction method.

Robust and globally efficient reduction of parametric, highly nonlinear computational models and real time online performance

This paper considers the problem of parametric, nonlinear, projection-based model order reduction (PMOR) and makes three contributions to advance its state of the art. First, it presents a computational strategy for reducing the total offline cost of nonlinear PMOR by enabling hyperreduction to be performed adaptively within the typical greedy sampling procedure, rather than uniformly at each of its iteration. Then, it presents a more accurate and computationally efficient approach for training hyperreduction based on residual Jacobians rather than residuals. Finally, it demonstrates for a large-scale, industrial-grade, parametric, nonlinear application, that the two aforementioned contributions make both offline and online stages of nonlinear PMOR practical in today’s engineering environments. The paper also shows that despite the so-called Kolmogorov n-width barrier for the model reduction of convection-dominated transport problems, the current state of the art of nonlinear, Petrov–Galerkin PMOR equipped with the advances proposed in this paper is capable of accelerating by orders of magnitude the accurate solution of shape-parametric, high Reynolds number, viscous flow problems represented by the Reynolds-averaged Navier–Stokes equations augmented with turbulence modeling. All three contributions are presented and discussed in the contexts of steady-state CFD problems; however, they are equally applicable to unsteady CFD problems as well as problems in solid mechanics and structural dynamics.