Galerkin Method Research Articles

MATHEMATICAL MODELING OF THE DYNAMICS OF THE AEROELASTIC "PIPELINE – PRESSURE SENSOR" SYSTEM

This paper proposes mathematical models of mechanical systems "pipeline - pressure sensor" designed to control the pressure of the working medium in the combustion chambers of engines. In such systems, to mitigate the impact of vibration accelerations and high temperatures, the sensor is connected to the engine using a pipeline and is located at some distance from it. The movement of the working medium is described by linear models of fluid and gas mechanics. To describe the dynamics of an elastic sensitive element, linear models of the mechanics of a solid deformable body are used. Based on linear differential equations with partial derivatives, mathematical formulations of problems are proposed that correspond to three-dimensional models of pressure measurement systems in gas-liquid media for some pipeline cross-sectional shapes, namely, for a pipeline with a rectangular cross-section, with a section in the form of a sector and in the form of a ring. By introducing integral characteristics, the solution of problems is reduced to studying one-dimensional models. Equations have been obtained that make it possible to determine the pressure of the working medium in the combustion chamber at each moment of time by the value of deformation of the sensitive element of the sensor. Analytical and numerical-analytical methods for solving the corresponding initial-boundary value problems for systems of differential equations are proposed. In the analytical approach, the solution of the problem is reduced to solving a differential equation with a deviating argument. The numerical-analytical study of the problem is based on the application of the Galerkin method. Also, a numerical experiment was carried out and examples of calculating the deformation of the sensitive element of the sensor in the case of rigid fastening when specifying specific values of the mechanical parameters of the system are presented, also during setting the law of change in the excess pressure of the working medium in the engine.

Stabilization of a weak viscoelastic wave equation in Riemannian geometric setting with an interior delay under nonlinear boundary dissipation

The stabilization of a weak viscoelastic wave equation with variable coefficients in the principal part of elliptic operator and an interior delay is considered. The dynamics is subject to a nonlinear boundary dissipation. This leads to a non-dissipative dynamics. The existence of solution is demonstrated by means of Faedo–Galerkin method combined with monotone operator theory in handling nonlinear boundary conditions. The main result pertains to exponential decay rates for energy, which depend on the geometry of the spatial domain, viscoelastic effects, the strength of delay and the strength of mechanical boundary damping. An important feature of the model is the fact that the delay term and stabilizing mechanism are not collocated geometrically — in contrast with many other works on the subject. This aspect of the problem requires the appropriate tools in order to exhibit propagation of the dissipation from one location to another. The precise ranges of admissible parameters characterizing the model and ensuring the stability are provided. The methods of proofs are routed in Riemannian geometry.

Highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method

The computation of electric field in transcranial magnetic stimulation (TMS) is essentially a problem of gradient calculation for thin layers. This paper introduces a hybrid-order hybridizable discontinuous Galerkin finite element method (HDG-FEM) and systematically demonstrates its superiority in TMS computations. The discrete format of HDG-FEM employing hybrid orders for TMS is derived and, from a fundamental numerical principle perspective, this study provides the elucidation of why HDG-FEM exhibits superior gradient computation capabilities compared to the widely used CG-FEM. Furthermore, the exceptional performance of HDG-FEM in thin layer calculation is demonstrated on both modified head models and realistic head models, focusing on three aspects: calculation errors, utilization of hybrid order, and computational cost. For the calculation of E-field in thin-layer regions with parameter mutation, the L∞ norm error of the first-order HDG-FEM with the same tetrahedral mesh is comparable to the L∞ norm error of the second-order CG-FEM. The L2 norm error of the same-order HDG-FEM is smaller than that of the same-order CG-FEM. By utilizing the hybrid order, HDG-FEM achieves a rapid reduction in errors of thin layers without a significant increase in the computational cost. This study transforms the three-dimensional TMS problem into a special two-dimensional problem for computation, reducing computational complexity from p3 in three dimensions to p2 in two dimensions, while achieving significantly higher accuracy compared to the commonly used CG-FEM. The utilization of hybrid orders in thin layers of the head demonstrates significant flexibility, making HDG-FEM a new alternative choice for TMS computations.

Stability and bifurcation analysis of simply supported rectangular nanoplate embedded on visco-Pasternak foundation based on first-order shear deformation theory

Vibrations of simply supported nanoplates embedded on visco-Pasternak foundation and stressed by in-plane periodic forces are investigated. The governing size-dependent equations employ the first-order shear deformation theory, nonlinear von Kármán strains, and the nonlocal elasticity theory. Reducing the governing system is carried out by the Bubnov-Galerkin method, which is based on two-mode model, thus the resulting system contains two ordinary differential equations with size- and time-dependent coefficients. Analysing the linearized system, the stability of the nanoplate structure is studied depending on the foundation parameters, force parameters and small-scale parameter. Nonlinear dynamics approaches are employed to determine the nature of vibration after the loss of stability. Time history, Poincare section, Lyapunov exponent are analysed for chosen values of excitation parameters, visco-Pasternak foundation parameters to indicate small-scale effects and nonlinear effects. The novelty of the work consists of the combined application of the nonlocal theory, first-order shear deformation theory, Floquet theory, methods of nonlinear dynamics and obtaining new results by a numerical experiment.

Convergence analysis of novel discontinuous Galerkin methods for a convection dominated problem

In this paper, we propose and analyze a numerically stable and convergent scheme for a convection-diffusion-reaction equation in the convection-dominated regime. Discontinuous Galerkin (DG) methods are considered since standard finite element methods for the convection-dominated equation cause spurious oscillations. We choose to follow a DG finite element differential calculus framework introduced in Feng et al. (2016) and approximate the infinite-dimensional operators in the equation with the finite-dimensional DG differential operators. Specifically, we construct the numerical method by using the dual-wind discontinuous Galerkin (DWDG) formulation for the diffusive term and the average discrete gradient operator for the convective term along with standard DG stabilization. We prove that the method converges optimally in the convection-dominated regime. Numerical results are provided to support the theoretical findings.

The Matrix‐Free Macro‐Element Hybridized Discontinuous Galerkin Method for Steady and Unsteady Compressible Flows

ABSTRACTThe macro‐element variant of the hybridized discontinuous Galerkin (HDG) method combines advantages of continuous and discontinuous finite element discretization. In this paper, we investigate the performance of the macro‐element HDG method for the analysis of compressible flow problems at moderate Reynolds numbers. To efficiently handle the corresponding large systems of equations, we explore several strategies at the solver level. On the one hand, we utilize a second‐layer static condensation approach that reduces the size of the local system matrix in each macro‐element and hence the factorization time of the local solver. On the other hand, we employ a multi‐level preconditioner based on the FGMRES solver for the global system that integrates well within a matrix‐free implementation. In addition, we integrate a standard diagonally implicit Runge–Kutta scheme for time integration. We test the matrix‐free macro‐element HDG method for compressible flow benchmarks, including Couette flow, flow past a sphere, and the Taylor–Green vortex. Our results show that unlike standard HDG, the macro‐element HDG method can operate efficiently for moderate polynomial degrees, as the local computational load can be flexibly increased via mesh refinement within a macro‐element. Our results also show that due to the balance of local and global operations, the reduction in degrees of freedom, and the reduction of the global problem size and the number of iterations for its solution, the macro‐element HDG method can be a competitive option for the analysis of compressible flow problems.

A divergence-free and [formula omitted]-conforming embedded-hybridized DG method for the incompressible resistive MHD equations

We present a divergence-free and Hdiv-conforming hybridized discontinuous Galerkin (HDG) method and a computationally efficient variant called embedded-HDG (E-HDG) for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. The proposed E-HDG approach uses continuous facet unknowns for the vector-valued solutions (velocity and magnetic fields) while it uses discontinuous facet unknowns for the scalar variable (pressure and magnetic pressure). This choice of function spaces makes E-HDG computationally far more advantageous, due to the much smaller number of degrees of freedom, compared to the HDG counterpart. The benefit is even more significant for three-dimensional/high-order/fine mesh scenarios. On simplicial meshes, the proposed methods with a specific choice of approximation spaces are well-posed for linear(ized) MHD equations. For nonlinear MHD problems, we present a simple approach exploiting the proposed linear discretizations by using a Picard iteration. The beauty of this approach is that the divergence-free and Hdiv-conforming properties of the velocity and magnetic fields are automatically carried over for nonlinear MHD equations. We study the accuracy and convergence of our E-HDG method for both linear and nonlinear MHD cases through various numerical experiments, including two- and three-dimensional problems with smooth and singular solutions. The numerical examples show that the proposed methods are pressure robust, and the divergence of the resulting velocity and magnetic fields is machine zero for both smooth and singular problems.

Control of a nonlinear wave equation with a dynamic boundary condition

Existence, uniqueness, energy decay, and approximate numerical solution for the nonlinear wave equation with dynamic control at the boundary is being studied in this work. The theoretical analysis of the problem will be conducted using the Faedo-Galerkin method and compactness results. To obtain the approximate numerical solution, a combined approach of the finite element method and a finite difference method will be employed, known as the linearized Crank-Nicolson Galerkin method. This method optimizes the calculations and preserves the quadratic order of convergence in time. Finally, numerical experiments are performed, and tables and graphs are presented to illustrate the theoretical convergence rates and demonstrate the consistency between theoretical and numerical results.

Optimal control of the gradient systems

In this paper we consider the optimal control problems governed by the gradient systems for the Ginzburg-Landau free energy where denotes a potential function and ε is the diffusivity. One example of gradient systems are the Schlögl equation arising in chemical waves with a quartic potential function F(y). Gradient systems are characterized by energy decreasing property . Numerical integrators that preserve the energy decreasing property in the discrete setting are called energy or gradient stable. It is known that the implicit Euler method is first order unconditionally energy stable method. The only second order unconditionally energy stable method is the average vector field (AVF) method. We discretize the gradients systems by discontinuous Galerkin method in space and by AVF integrator in time. We solve optimal control problems for the Schlögl equation with traveling and spiraling waves using sparse and regularized controls .

The a posteriori error estimates and an adaptive algorithm of the discontinuous Galerkin method for the modified transmission eigenvalue problem with absorbing media

The a posteriori error estimates and an adaptive algorithm of the discontinuous Galerkin method for the modified transmission eigenvalue problem with absorbing media

Numerical Solution of Differential Equations Using the Wavelet-Based Galerkin Method with Fibonacci Wavelets

Differential equations form the foundation of scientific theories that address numerous real-world physical challenges. Numerical methods enable the resolution of complex problems through relatively simple operations. A significant advantage of numerical methods, compared to analytical methods, is their ease of implementation on modern computers, allowing for quicker solutions. Galerkin's method belongs to a broader category of numerical techniques. Additionally, wavelet analysis represents a promising domain within applied and computational research. This paper establishes a wavelet-based Galerkin method for numerically solving differential equations, utilizing Fibonacci wavelets as trial functions. The proposed method yields results comparable to existing techniques and provides solutions that closely approximate exact answers for certain problems, thereby demonstrating its effectiveness and accuracy.

Topology Optimization for Elasticity Problem of Isotropic and Orthotropic Materials Based on the Complex Variable Element-Free Galerkin Method

By using the improved complex variable moving least squares (ICVMLS) approximation to construct the shape function, thus the improved complex variable element-free Galerkin (ICVEFG) method is established for analyzing two types of topology optimization problems including isotropic and orthotropic materials in this study. Solid isotropic microstructure with penalization (SIMP) is utilized with relative density of nodes selected as the design variable, volume fraction set as the constraint and structural flexibility selected as the objective function, thus a mathematical model for topology optimization is developed, and the formula of sensitivity is derived. Four optimization examples of isotropic and orthotropic materials are given, and the feasibility of the ICVEFG method for solving topology optimization problem is demonstrated, in addition, the checkerboard phenomenon can be avoided; it is also shown that the ICVEFG method is more advantageous than the EFG method in solving orthotropic beam as well as isotropic problem with two fixed ends.

A Discrete Sine‐Cosine Transforms Galerkin Method for the Conductivity of Heterogeneous Materials With Mixed Dirichlet/Neumann Boundary Conditions

ABSTRACTThis work aims at developing a numerical method for conductivity problems in heterogeneous media subjected to mixed Dirichlet/Neumann boundary conditions. The method relies on a fixed‐point iterative solution of an auxiliary problem obtained by a Galerkin discretization using an approximation space spanned by mixed cosine‐sine series. The solution field is written as a known term verifying the boundary conditions and an unknown term described by cosine‐sine series, having no contribution on the boundary. Discrete sine‐cosine transforms, of Type I and III depending on the boundary conditions, are used to approximate the elementary integrals involved in the Galerkin formulation, which makes the method relying on the numerical complexity of fast Fourier transforms. The method is finally assessed in a problem of a composite material.

The simplified weak Galerkin method with θ scheme and its reduced-order model for the elastodynamic problem on polygonal mesh

This paper presents a simplified weak Galerkin (SWG) method for solving the elastodynamic problem and its reduced-order model (ROM) using the proper orthogonal decomposition (POD) technique. The SWG method allows for the use of polygonal meshes. It only utilizes degrees of freedom associated with the boundary, reducing computational complexity compared to the classical weak Galerkin method. Moreover, we apply the POD technique to develop a POD-SWG-ROM for the problem, further enhancing the computational efficiency. Then, to discretize in time, we utilize a θ-scheme, where the scheme is explicit when 0≤θ<1/4 and implicit when 1/4≤θ≤1/2. We establish the theoretical analysis of the semi-discrete scheme and the fully-discrete θ scheme. The theoretical analysis demonstrates that the method is locking-free, and the convergence rate in the H1 and L2 norms is O(Δt2+h1) and O(Δt2+h2) respectively. Finally, we verify the theoretical analysis through numerical tests and effectively simulate the propagation of elastic waves under polygonal meshes. Moreover, the proposed POD-SWG-ROM can significantly improve computational efficiency.

A Physics‐Driven GraphSAGE Method for Physical Field Simulations Described by Partial Differential Equations

AbstractPhysics‐informed neural networks (PINNs) have successfully addressed various computational physics problems based on partial differential equations (PDEs). However, while tackling issues related to irregularities like singularities and oscillations, trained solutions usually suffer low accuracy. In addition, most current works only offer the trained solution for predetermined input parameters. If any change occurs in input parameters, transfer learning or retraining is required, and traditional numerical techniques also need recomputation. In this work, a physics‐driven GraphSAGE approach (PD‐GraphSAGE) based on the Galerkin method and piecewise polynomial nodal basis functions is presented to solve computational problems governed by irregular PDEs and to develop parametric PDE surrogate models. This approach employs graph representations of physical domains, thereby reducing the demands for evaluated points due to local refinement. A distance‐related edge feature and a feature mapping strategy are devised to help training and convergence for singularity and oscillation situations, respectively. The merits of the proposed method are demonstrated through a couple of cases. Moreover, the robust PDE surrogate model for heat conduction problems parameterized by the Gaussian random field source is successfully established, which not only provides the solution accurately but is several times faster than the finite element method in the experiments.

Accurate Closed-Form Solutions for the Free Vibration and Supersonic Flutter of Laminated Circular Cylindrical Shells

According to the Donnell–Mushtari shell theory, this work presents a closed-form solution procedure for free vibration of open laminated circular cylindrical shells with arbitrary homogeneous boundary conditions (BCs). The governing differential equations of free vibration are derived from the Rayleigh quotient and solved by the iterative separation-of-variable (iSOV) method. In addition, considering axial aerodynamic pressure, simulated by the linear piston theory, the exact eigensolutions for the flutter of open laminated cylindrical shells with simply supported circumferential edges and closed laminated cylindrical shells are also achieved. The governing differential equations of cylindrical shell flutter are derived from the Hamilton variational principle and solved by the separation-of-variable (SOV) method. The influence of circumferential dimension on flutter speed is investigated for open cylindrical shells, which reveals that the number of circumferential waves in critical flutter mode increases with circumferential length, and there exists an infimum for flutter speed that is an invariant independent of circumferential length. The present results agree well with those obtained by the Galerkin method, the finite element method, and other analytical methods.

An approximate solution of multi-term fractional telegraph equation with quadratic B-spline basis functions

This paper introduces the Galerkin Method for the approximate solution ofMulti-term Fractional Telegraph Equations (MFTE). The Galerkin Method (GM)is one of the most popular techniques for the solution of Partial Differential Equations (PDEs), and it uses the idea of mapping a solution onto a set of basis functions and then seeking the residual error through minimization. In GM, the weight and basis functions are the same, and as such, the basis functions are selected appropriately to satisfy the given conditions imposed. In this paper, the quadratic B-spline functions are adopted as shape and test functions for resolving the approximate solution of MFTE. Here, the Caputo fractional derivative takes care of the fractional part, and the Gauss-Mamadu-Njoseh quadrature scheme handles numerical integration. Numerical illustrations are examined for single-term and two-term MFTE, respectively, with numerical evidence measured usingL2 and L∞ error norms. Consequently, the resulting numerical evidence, as presented in Tables and Figures, shows the accuracy and reliability of the method. Also, the study examined and presented relevant theorems of convergence and error analyses of method. MAPLE 18 was used for all computational frameworks in this research.

Towards the Galerkin approximation of tetraskelion metamaterials

Towards the Galerkin approximation of tetraskelion metamaterials

Analysis of the Staggered DG Method for the Quasi-Newtonian Stokes flows

AbstractThis paper introduces and analyzes a staggered discontinuous Galerkin (DG) method for quasi-Newtonian Stokes flow problems on polytopal meshes. The method introduces the flux and tensor gradient of the velocity as additional unknowns and eliminates the pressure variable via the incompressibility condition. Thanks to the subtle construction of the finite element spaces used in our staggered DG method, no additional numerical flux or stabilization terms are needed. Based on the abstract theory for the non-linear twofold saddle point problems, we prove the well-posedness of our scheme. A priori error analysis for all the involved unknowns is also provided. In addition, the proposed scheme can be hybridizable and the global problem only involves the trace variables, rendering the method computationally attractive. Finally, several numerical experiments are carried out to illustrate the performance of our scheme.

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

AbstractThe proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.