Legendre Spectral Element Method Research Articles

A Legendre Spectral-Element Method to Incorporate Topography for 2.5D Direct-Current-Resistivity Forward Modeling

An effective and accurate solver for the direct-current-resistivity forward-modeling problem has become a cutting-edge research topic. However, computational limitations arise due to the substantial amount of data involved, hindering the widespread use of three-dimensional forward modeling, which is otherwise considered the most effective approach for identifying geo-electrical anomalies. An efficient compromise, or potentially an alternative, is found in two-and-a-half-dimensional (2.5D) modeling, which employs a three-dimensional current source within a two-dimensional subsurface medium. Consequently, a Legendre spectral-element algorithm is developed specifically for 2.5D direct-current-resistivity forward modeling, taking into account the presence of topography. This numerical algorithm can combine the complex geometric flexibility of the finite-element method with the high precision of the spectral method. To solve the wavenumber-domain electrical potential variational problem, which is converted into the two-dimensional Helmholtz equation with mixed boundary conditions, the Gauss–Lobatto–Legendre (GLL) quadrature is employed in all discrete quadrilateral spectral elements, ensuring identical Legendre polynomial interpolation and quadrature points. The Legendre spectral-element method is applied to solve a two-dimensional Helmholtz equation and a resistivity half-space model. Numerical experiments demonstrate that the proposed approach yields highly accurate numerical results, even with a coarse mesh. Additionally, the Legendre spectral-element algorithm is employed to simulate the apparent resistivity distortions caused by surface topographical variations in the direct-current resistivity Wenner-alpha array. These numerical results affirm the substantial impact of topographical variations on the apparent resistivity data obtained in the field. Consequently, when interpreting field data, it is crucial to consider topographic effects to the extent they can be simulated. Moreover, our numerical method can be extended and implemented for a more accurate computation of three-dimensional direct-current-resistivity forward modeling.

Efficient spectral and spectral element methods for Sobolev equation with diagonalization technique

In this paper, we first introduce a new series of Legendre basis functions by using the matrix decomposition technique, which are simultaneously orthogonal in both L2- and H1-inner products. Then, we construct efficient space-time spectral method for linear Sobolev equation using spectral approximation in space and multi-domain collocation approximation in time, which can be implemented in a synchronous parallel fashion. Next, we propose a novel Legendre spectral element method for solving nonlinear Sobolev equation with Burgers' type term, which reduce the non-zero entries of linear systems and computational cost. Some rigorous error estimates are carried out for one-dimensional Sobolev equation. Numerical experiments illustrate the effectiveness and accuracy of the suggested approaches.

A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition

A curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems.

Linear stability analysis on the thermocapillary flow of molten silicon in a liquid bridge between unequal disks under a rotating magnetic field

This paper explores the stability of thermocapillary flow in a molten silicon liquid bridge situated between two unequal disks and subjected to a rotating magnetic field. Two different heating strategies are examined, and the static deformation of the free surface is taken into account by solving the Young–Laplace equation. Using linear stability analysis based on the Legendre spectral element method, the critical conditions are determined, and the underlying instability mechanism is uncovered through energy analysis. When heated from the bottom disk, three transitions between axisymmetric steady flow and three-dimensional oscillatory flow are observed within a narrow range of 7000 ≤ Ta ≤ 9000. In contrast, when heated from the top disk, the rotating magnetic field significantly enhances the flow stability. Furthermore, based on the energy analysis, we identify two types of instability caused by purely hydrodynamic mechanisms.

A high-order fluid–structure interaction framework with application to shock-wave/turbulent boundary-layer interaction over an elastic panel

Within this work, a loosely-coupled high-order fluid–structure interaction (FSI) framework is developed in order to investigate the influence of an elastic panel response on shock-wave/turbulent boundary-layer interaction (SWTBLI). Since high-order methods are expected to determine the future of high-fidelity numerical simulations, they are employed in the construction of both fluid and structure solvers. Specifically, a split-form arbitrary Lagrangian-Eulerian discontinuous Galerkin spectral element method is employed in the fluid solver and a Legendre spectral finite element method in the structure solver. Shock capturing by an improved adaptive filter method, which confines the filtering effect to the vicinity of shocks, is found to be well-behaved in accuracy, efficiency and flexibility.After being validated by two benchmark FSI problems, the developed FSI framework is applied to simulate SWTBLI over an elastic panel. A comparison with a previous simulation of SWTBLI over a rigid panel reveals that: (1) A larger amplitude of the pressure variation, observed on the elastic panel surface, implies a larger threat to the structural integrity; (2) The shock-induced separation flow over the elastic panel changes both in size and shape, leading to a different skin-friction coefficient distribution; (3) A new low-frequency flow unsteadiness of the same magnitude as the elastic panel vibration is detected, which may affect the flow dynamics inside the turbulent boundary layer.

Effect of rotating magnetic field on the stability of thermocapillary flow in a gallium arsenide liquid bridge between unequal ends

In this study, we investigated the impact of a rotating magnetic field on the stability of a thermocapillary flow in a gallium arsenide liquid bridge (Prandtl number Pr = 0.068) situated between two unequal disks, considering two different scenarios with radius ratios of Γr = 0.98 and Γr = 0.60 for the upper heated disk. By utilizing linear stability analysis based on the Legendre spectral element method, we first identified the critical parameters of the onset of flow instability, including critical Marangoni number (Mac), dimensionless oscillation frequency (fc), and azimuthal wavenumber (m). Then, we employed kinetic energy budget analysis to uncover the underlying instability mechanism. For radius ratio Γr = 0.98, three transitions between axisymmetric steady flow and three-dimensional oscillatory flow in the narrow range of Taylor number Ta (8700≤Ta ≤ 9500) are observed; these transitions arise due to the interplay between the flow induced by rotating magnetic field and thermocapillary flow. For the Γr = 0.60 scenario, the rotating magnetic field is observed to significantly enhance the flow stability. Additionally, our analysis identifies four instability types dominated by the hydrodynamic mechanism. In the meantime, the thermocapillary mechanism also contributes to flow instability in the specific region of Taylor number Ta (1250≤Ta ≤ 8000) for radius ratio Γr = 0.98.

Using Legendre Spectral Element Method with Split Bregman Method for Solving Optimal Control Problem with Elliptic Partial Differential Equation Constraint

Using Legendre Spectral Element Method with Split Bregman Method for Solving Optimal Control Problem with Elliptic Partial Differential Equation Constraint

Instability mechanisms of thermocapillary liquid bridges between disks of unequal radii

In this paper, we explore thermocapillary liquid bridges between two disks of unequal radii with Prandtl numbers Pr of 0.0258 (mercury) and 0.068 (gallium arsenide) to gain insight into the underlying instability mechanism. In the context of Legendre's spectral element method, we determine critical conditions via linear stability analysis and then identify the instability mechanism through energy analysis. For the mercury bridge (Pr = 0.0258), our analysis suggests that the flow instability undergoes an oscillatory bifurcation for radius ratios in the range of 0.5 ≤ Γr ≤ 0.66. In particular, we found three transitions between two-dimensional steady axisymmetric flow and three-dimensional stationary flow by further increasing the radius ratio to 0.73 ≤ Γr ≤ 0.76. For the gallium arsenide liquid bridge (Pr = 0.068), the instability is always an oscillatory bifurcation in the whole computational interval. Furthermore, our observations identify six instability modes with different mechanisms. All instability modes in the mercury bridge (Pr = 0.0258) are purely hydrodynamic, but the thermocapillary mechanism cannot be ignored in the gallium arsenide liquid bridge (Pr = 0.068) because of the enhanced Pr effect.

A Legendre spectral element method for the family of regularized long wave equations

Regularized Long Wave (RLW) equation is one of the most important nonlinear PDEs. It is related to the famous KdV equation and acts as an alternative for scrutinizing soliton phenomena. In this paper, we use the Legendre Spectral Element Method (LSEM) for the numerical study of the family of the RLW equations containing RLW, modified RLW, and generalized RLW. We perform the space discretization by the LSEM, and the Crank–Nicolson method is applied to discretize the time. To confirm the accuracy and efficiency of the proposed method, the conservation properties of the equations are checked, and various types of errors are reported.

Flow instabilities in thermocapillary liquid bridges between two coaxial disks with different radii

The primary instability of thermocapillary flow in a liquid bridge between two coaxial disks with different radii is investigated under microgravity for silicon melt with Prandtl number Pr=0.011 via two heating strategies. The static deformation of the free surface is considered by solving the Young-Laplace equation. The physical instability mechanisms are explored by analyzing the energy budgets of the critical modes, which are determined by linear stability analysis based on the Legendre spectral element method. With the decrease of radius ratio Γr, the stability of thermocapillary flow is significantly improved. In contrast to typical cylindrical liquid bridges (Γr=1), the instability is an oscillatory bifurcation for small radius ratios (Γr≤0.672) when the liquid bridge is heated from the bottom disk. Furthermore, the instabilities for all radius ratios and heating strategies are found to be purely hydrodynamic, but the specific instability mechanisms are different.

Numerical investigations on model order reduction to SEM based on POD-DEIM to linear/nonlinear heat transfer problems

The motivation in this article is to foster and conduct an in-depth investigation on reduced-order modeling (ROM) techniques via proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) in conjunction with high-order Legendre spectral element method (SEM) applied for time-dependent linear and nonlinear heat conduction problems. This approach significantly relieves the CPU burden, yet preserves the numerical accuracy of high-order schemes; this is especially pronounced for linear transient problems. For nonlinear cases, we use DEIM to save CPU time for nonlinear counterparts. For this purpose, we first obtain training data by solving the system of full-order models (FOM) using snapshot techniques to construct the POD basis. The well-known generalized single step single solve (GSSSS-1) framework is next employed for the first-order system for the time discretization. Numerical results for both linear and nonlinear heat conduction problems such as in a nuclear reactor, etc., show excellent agreement between FOM solutions and ROM solutions, and the CPU advantages via POD ROM techniques are also prominent for high-order Legendre SEM.

Numerical solution of two and three dimensional time fractional damped nonlinear Klein–Gordon equation using ADI spectral element method

At the present work, we propose a new numerical scheme for the solution of two and three dimensional time fractional nonlinear damped Klein–Gordon equation (DKGE). To this end, we use the Legendre spectral element method to discretize the equation in the spatial directions and for the time stepping, an alternating direction implicit (ADI) method based on a scheme of order O(τ2) is considered. We prove that the presented method is unconditionally stable and obtain an error estimate for the fully discrete scheme. The presented numerical experiments demonstrate the good accuracy and efficiency of the proposed ADI spectral element method.

An efficient numerical method for the solution of 2D variable order time fractional mobile–immobile advection–dispersion model

In the literature, there are a few efficient numerical methods with stability and convergence analysis for the solution of variable order fractional partial differential equations in higher dimensions. In this work, we propose an efficient numerical method for the solution of two‐dimensional variable order time fractional mobile–immobile advection–dispersion model with spatially variable convection coefficient. At first, we discretize the time derivatives with a second‐order scheme. Then, we apply the Legendre spectral element method in spatial directions and obtain the fully discrete scheme. We prove that the semi‐discrete scheme is unconditionally stable and present an error estimate for the fully discrete scheme. The numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

Spectral Element Methods a Priori and a Posteriori Error Estimates for Penalized Unilateral Obstacle Problem

The purpose of this paper is the determination of the numerical solution of a classical unilateral stationary elliptic obstacle problem. The numerical technique combines Moreau-Yoshida penalty and spectral finite element approximations. The penalized method transforms the obstacle problem into a family of semilinear partial differential equations. The discretization uses a non-overlapping spectral finite element method with Legendre–Gauss–Lobatto nodal basis using a conforming mesh. The strategy is based on approximating the solution using a spectral finite element method. In addition, by coupling the penalty and the discretization parameters, we prove a priori and a posteriori error estimates where reliability and efficiency of the estimators are shown for Legendre spectral finite element method. Such estimators can be used to construct adaptive methods for obstacle problems. Moreover, numerical results are given to corroborate our error estimates.

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

AbstractThis article is devoted to solving numerically the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi‐discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction isO(dt). Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss–Legendre–Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the one‐ and two‐dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples.

Dynamical behavior of reaction–diffusion neural networks and their synchronization arising in modeling epileptic seizure: A numerical simulation study

Excessive synchronizations of neurons in the brain networks can be a reason for some episodic disorders such as epilepsy. In this paper, we simulate neural dynamic models and their synchronizations numerically by a new numerical algorithm. In order to overcome the complexity of the problem, a suitable combination of the Legendre spectral element method and operator splitting technique is proposed to convert a complex system of partial differential equations into sparse linear algebraic systems. The simulations are explored in different aspects such as accuracy, computing the wavefront and angular speeds, computing time, the condition number of obtained algebraic systems to demonstrate the proposed scheme effectiveness. The numerical simulations and comparisons confirm the method is suitable in terms of accuracy and speed.

Legendre spectral element method (LSEM) to simulate the two-dimensional system of nonlinear stochastic advection–reaction–diffusion models

In this work, we develop a Legendre spectral element method (LSEM) for solving the stochastic nonlinear system of advection–reaction–diffusion models. The used basis functions are based on a class of Legendre functions such that their mass and diffuse matrices are tridiagonal and diagonal, respectively. The temporal variable is discretized by a Crank–Nicolson finite-difference formulation. In the stochastic direction, we also employ a random variable W based on the Q-Wiener process. We inspect the rate of convergence and the unconditional stability for the achieved semi-discrete formulation. Then, the Legendre spectral element technique is used to obtain a full-discrete scheme. The error estimation of the proposed numerical scheme is substantiated based upon the energy method. The numerical results confirm the theoretical analysis.

Application of spectral element method for solving Sobolev equations with error estimation

This paper is dedicated to numerically solving the Sobolev equations that have several applications in physics and mechanical engineering. First, the temporal derivative is discretized by the Crank-Nicolson finite difference technique to obtain a semi-discrete scheme in the temporal direction. Afterward, the stability and convergence analysis of the time semi-discrete scheme are proven by applying the energy method. It also implies that the convergence order in the temporal direction is O(dt2). Second, a fully discrete formula has been acquired by discretizing the spatial derivatives via Legendre spectral element method (LSEM). This method applies the Lagrange polynomial based on the Gauss-Legendre-Lobatto (GLL) points. Moreover, an error estimation is given for the obtained fully discrete scheme. Eventually, the two-dimensional Sobolev equations are solved by using the proposed procedure. The accuracy and efficiency of the mentioned procedure are demonstrated by several numerical examples.

Legendre spectral element method for solving Volterra-integro differential equations

In this article, we present a spectral element method for numerical solution of linear Volterra integro-differential equations with boundary conditions. First, we obtain discrete form of the problem using the spectral element method. We use the Legendre polynomials for interpolation. Also, the error analysis of the method is considered. Furthermore, the efficiency of the proposed method will be considered through numerical examples.

Convergence analysis of an efficient spectral element method for Stokes eigenvalue problem

In this work, an efficient Legendre spectral element method was proposed to solve the two‐dimensional Stokes eigenvalue problem on L‐shaped domain. Based on minimax principle, the rigorous error estimates of the approximate eigenvalues are proved. The approximate eigenvalues converge with the order in L2 norm, where h,N, and s are space grid size, polynomial degree, and the regularity of exact solution, respectively. Several numerical examples are provided to verify the theoretical analysis and demonstrate the effectiveness of the proposed scheme.