Three-dimensional complex variable element-free Galerkin method

The complex variable element-free Galerkin (CVEFG) method is an efficient meshless Galerkin method that uses the complex variable moving least squares (CVMLS) approximation to form shape functions. In the past, applications of the CVMLS approximation and the CVEFG method are confined to 2D problems. This paper is devoted to 3D problems. Computational formulas and theoretical analysis of the CVMLS approximation on 3D domains are developed. The approximation of a 3D function is formed with 2D basis functions. Compared with the moving least squares approximation, the CVMLS approximation involves fewer coefficients and thus consumes less computing times. Formulations and error analysis of the CVEFG method to 3D elliptic problems and 3D wave equations are provided. Numerical examples are given to verify the convergence and accuracy of the method. Numerical results reveal that the CVEFG method has better accuracy and higher computational efficiency than other methods such as the element-free Galerkin method.     

GFV solution on UTE mesh for transient modeling of concrete aging effects on thermal plane strains during construction of gravity dam

Considering variation of the material properties during early ages of construction, the strain–stress fields due to cement hydration induced transient temperature field in a section of a concrete dam are computed. The 2D matrix free Galerkin Finite Volume (GFV) method is utilized to solve temperature and plane-strain equations in a sequential manner on an Unstructured Triangular Elements (UTE) meshes. For the computed temperature field at each time step some iterative solutions are performed until the force equilibrium equations are converged. In the developed numerical model, a novel method to impose gradient boundary condition is introduced that is suitable for the GFV solution on UTE meshes. The results of the developed model is verified by comparing the computed results with the experimental measurements of some bench mark test cases, and good agreement is observed. To present the ability of the introduced model to model real problems, modeling time dependent thermal stress profiles during gradual construction of a concrete dam on a natural foundation is performed for both constant and variable mechanical properties conditions.     

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

In order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled with $h$-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.     

Analysis of stress and strain in tension cylindrical sample

One of the basic engineering problem occurring during the numerical analysis is to define the function of yield stress of material in the real conditions of a technological process. These properties are necessary to calculate the deformation and the state of stress and strain in the surface layer of an object. An inappropriate selection of the mechanical properties of the material is the reason of the occurrence of errors in numerical calculations of a continuous object, considered as a boundary and initial problem. Scientific investigations are being conducted with the aim to develop a database concerning yield stresses for different metals, depends on complex conditions of thermo-dynamical loads, e.g. temperature, the equivalent of the strain and the strain rate. The article presents a method of the determination of this dependence while using an experimental and numerical analysis. During the model investigations on the INSTRON testing machine, the force of elongation of the sample is measured and then calculations are made of the displacement of nodes of finite elements, plotted on outside surface of sample. The process is considered as a multi nonlinear problem. For this reason, an incremental method of motion and deformation of solid in an updated Lagrange formulation is used. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary

We present a discontinuous Galerkin method, based on the classical method of Nitsche, for elliptic problems with an immersed boundary representation on a structured grid. In such methods very small elements typically occur at the boundary, leading to breakdown of the discrete coercivity as well as numerical instabilities. In this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain. This construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates. Furthermore, we prove a bound of the condition number of the stiffness matrix. All the results are valid for polynomials of arbitrary order. We also discuss the implementation of the method and present numerical examples in three dimensions.     

Three-dimensional simulation of flat rolling through a combined finite element–boundary element approach

This paper presents an innovative approach for analysing three-dimensional flat rolling. The proposed approach is based on a solution resulting from the combination of the finite element method with the boundary element method. The finite element method is used to perform the rigid–plastic numerical modelling of the workpiece allowing the estimation of the roll separating force, rolling torque and contact pressure along the surface of the rolls. The boundary element method is applied for computing the elastic deformation of the rolls. The combination of the two numerical methods is made using the finite element solution of the contact pressure along the surface of the rolls to define the boundary conditions to be applied on the elastic analysis of the rolls. The validity of the proposed approach is discussed by comparing the theoretical predictions with experimental data found in the literature.     

On magnetoelastic problems of a plane with an arbitrarily shaped hole under stress and displacement boundary conditions

An analytical method for the static plane problem of magnetoelasticityis developed for an infinite plane containing a hole of arbitraryshape under stress and displacement boundary conditions in aprimary uniform magnetic field. The magnetic field influencesthe elastic field by introducing a body force called the Lorentzponderomotive force in the equilibrium equations. The body forcecan be further described in a form relating with the electromagneticstress tensor. The complex variable method in conjunction withthe rational mapping function technique is used in the analysisfor both magnetic field and mechanical field. Governing equationsand boundary conditions are expressed in terms of complex functions.Complex magnetic potential and stress functions are obtainedusing Cauchy integrals for the paramagnetic and soft ferromagneticmaterials, respectively. The distributions of magnetic fieldand the stress components are shown for certain directions ofprimary magnetic fields in an infinite plane with a square hole,as an example. It is found that the stress distributions forthe two types of materials are identical despite the differenceof magnetic fields. The extreme cases of a free and a fixedhole reduced to a crack and a rigid fibre, respectively, arealso investigated. The stress intensity factors at the tipsof crack and rigid fibre are computed, and their variation forcertain directions of primary magnetic field is shown.     

Secondary Buckling Analysis of Thin Rectangular Plates Based on the Wavelet Galerkin Method北大核心CSCD

Application of the wavelet Galerkin method (WGM) to numerical solution of nonlinear buckling problems was studied with classical elastic thin rectangular plates. First, the discretized scheme of the von Kármán equation were introduced, then a simple calculation approach to the Jacobian and Hessian matrices based on the WGM was proposed, and the wavelet discretized scheme-based eigenvalue equation method, the extended equation method and the pseudo arc-length method for nonlinear buckling analysis were discussed. Second, the secondary post-buckling equilibrium paths of elastic thin rectangular plates and the effects of aspect ratios, boundary conditions and bi-directional compression on the mode jumping behaviors, were discussed in detail. Numerical results show that, the WGM possesses good convergence for solving buckling loads on rectangular plates, and the obtained equilibrium paths are in good agreement with those from the stability experiments, the 2-step perturbation method and the nonlinear finite element method. Given the feasibility of combination with different bifurcation computation methods, the WGM makes an efficient spatial discretization method for complex nonlinear stability problems of typical plates and shells. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

The fully Sinc‐Galerkin method for time‐dependent boundary conditions

The fully Sinc‐Galerkin method is developed for a family of complex‐valued partial differential equations with time‐dependent boundary conditions. The Sinc‐Galerkin discrete system is formulated and represented by a Kronecker product form of those equations. The numerical solution is efficiently calculated and the method exhibits an exponential convergence rate. Several examples, some with a real‐valued solution and some with a complex‐valued solution, are used to demonstrate the performance of this method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

功能梯度梁在热-机械荷载作用下的几何非线性分析

基于经典梁理论，运用虚功原理和变分法推导了均匀变温场与横向均布荷载联合作用的功能梯度梁的几何非线性控制方程.考虑端部不可移夹紧边界条件，运用打靶法求解了该两点边值问题.当横向均布荷载为0时，考察了功能梯度梁的热屈曲临界升温和屈曲平衡路径.当均匀变温与横向均布荷载都不为0时，考察了功能梯度梁的荷载 挠度曲线.数值结果表明:随材料体积分数指数增加，梁的有量纲热屈曲临界升温显著减小，后屈曲变形显著增加；变温对功能梯度梁的荷载 挠度曲线影响非常显著.发现了功能梯度梁的双稳态构形及其转换现象，梁的最终平衡位形不但与变温及荷载参数有关，还与加载历程有关.     

新的三维力学GELD正演和反演算法

在本文中 ,我们提出了新的整体积分和局部微分GILD的力学正演和反演方法 .我们建立了弹性和塑性力学的体积分微分方程 .我们证明了这个体积分方程和伽辽金虚功原理等价 .新的GILD方法是基于这个体积分微分方程 .GL方法是进一步的发展 ,GL是一种整体场和局部场相互作用的全新方法 .在这个方法中 ,仅仅需要解 3× 3或者 6 × 6的局部小矩阵 .特别是 ,用GL方法求解无限域的偏微分方程时 ,不需要任何人工边界 ,不需要任何吸收边界条件和不需要任何边界积分方程 .新的三维力学GILD正演和反演算法已被应用研究奈米材料的力学性质的模拟计算 .我们获得非常好的奈米材料的力学变形的超拉力的力学性质 .我们提出了新的奈米地球物理新概念和发现了GILD数值量子     

A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions

Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penalty method. In the work, we conduct a comparison study for elliptic problems with four different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkin method and deep Ritz method. In the former, the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions.Besides, in some cases, when the boundary condition can be implemented in an exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process.     

粘性流体两相多孔介质非线性动力问题的罚有限元法

采用基于混合物理论的多孔介质模型,给出粘性流体饱和两相多孔介质非线性动力问题的控制场方程以及相应边值和初值问题的提法,用Galerkin加权残值法导出罚有限元公式,并给出该非线性方程组的迭代求解方法。考虑了体积分数和渗透率与变形相关的情况。用编制的有限元程序计算分析了一维多孔柱体在脉冲载荷作用下的瞬态响应,数值结果表明文中方法正确有效。     

变厚度圆薄板非对称非线性弯曲问题

首先将直角坐标系中的横向变厚度薄板的大挠度方程,转化到极坐标系中的变厚度圆薄板的非对称大挠度方程·　此方程和极坐标系中径向、切向两个平衡方程联立求解·　将物理方程和中面应变非线性变形方程,代入3个平衡方程,可得用3个变形位移表示的3个非对称非线性方程·　用Fourier级数表示的解代入基本方程,获得相应的基本方程·　在周边夹紧边界条件下,用修正迭代法求解·　作为算例,研究了余弦形式载荷作用下的问题,还给出了载荷与挠度的特征曲线,曲线依据变厚度参数变化而变化,其结果和物理概念完全吻合·     

Imposing boundary conditions in Sinc method using highest derivative approximation

Sinc approximate methods are often used to solve complex boundary value problems such as problems on unbounded domains or problems with endpoint singularities. A recent implementation of the Sinc method [Li, C. and Wu, X., Numerical solution of differential equations using Sinc method based on the interpolation of the highest derivatives, Applied Mathematical Modeling 31 (1) 2007 1–9] in which Sinc basis functions are used to approximate the highest derivative in the governing equation of the boundary value problem is evaluated for structural mechanics applications in which interlaminar stresses are desired. We suggest an alternative approach for specifying the boundary conditions, and we compare the numerical results for analysis of a laminated composite Timoshenko beam, implementing both Li and Wu’s approach and our alternative approach for applying the boundary conditions. For the Timoshenko beam problem, we obtain accurate results using both approaches, including transverse shear stress by integration of the 3D equilibrium equations of elasticity. The beam results indicate our approach is less dependent on the selection of the Sinc mesh size than Li and Wu’s SIHD. We also apply SIHD to analyze a classical laminated composite plate. For the plate example, we experience difficulty in obtaining a complete system of equations using Li and Wu’s approach. For our approach, we suggest that additional necessary information may be obtained by applying the derivatives of the boundary conditions on each edge. Using this technique, we obtain accurate results for deflection and stresses, including interlaminar stresses by integration of the 3D equilibrium equations of elasticity. Our results for both the beam and the plate problems indicate that this approach is easily implemented, has a high level of accuracy, and good convergence properties.     

Hybrid numerical method with adaptive overlapping meshes for solving nonstationary problems in continuum mechanics

Techniques that improve the accuracy of numerical solutions and reduce their computational costs are discussed as applied to continuum mechanics problems with complex time-varying geometry. The approach combines shock-capturing computations with the following methods: (1) overlapping meshes for specifying complex geometry; (2) elastic arbitrarily moving adaptive meshes for minimizing the approximation errors near shock waves, boundary layers, contact discontinuities, and moving boundaries; (3) matrix-free implementation of efficient iterative and explicit–implicit finite element schemes; (4) balancing viscosity (version of the stabilized Petrov–Galerkin method); (5) exponential adjustment of physical viscosity coefficients; and (6) stepwise correction of solutions for providing their monotonicity and conservativeness.     

An iterative approach to solve a nonlinear moving boundary problem describing the solvent diffusion within glassy polymers

This paper consists in studying a mathematical model of solvent diffusion through the glassy polymers as a one-dimensional moving boundary problem with kinetic undercooling. We establish an iterative variable time-step method based on a nonstandard finite difference (NSFD) scheme to solve the considered moving boundary problem. The monotonicity and positivity of the numerical solution are proved. The numerical approach is investigated for three test problems composed of constant and inconstant diffusion coefficients for different values of parameters to demonstrate the validity and ability of the method.     

Modelling of the deformation processes and the localization of plastic deformations in the torsion–tension of solids of revolution

Generalized two-dimensional problems of the torsion of elastoplastic solids of revolution of arbitrary shape for large deformations under non-uniform stress-strain conditions are formulated and a method for their numerical solution is proposed. The use of this method to construct strain diagrams of materials based on experiments on the torsion of axisysmmetric samples of variable thickness until fracture occurs is described. Experimental and numerical investigations of processes of elastoplastic deformation, loss of stability and supercritical behaviour of solid cylindrical steel samples of variable thickness under conditions of monotonic kinematic loading with a torque, a tension and a combined load are presented. The mutual influence of torsion and tension on the deformation process and the limit states is estimated, and the universality (the independence of the form of the stress-strain state) of the “stress intensity – Odqvist parameter” diagram for steel for large deformations is proved.     

A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions

We consider retarded boundary integral formulations of the three-dimensional wave equation in unbounded domains. Our goal is to apply a Galerkin method in space and time in order to solve these problems numerically. In this approach the computation of the system matrix entries is the major bottleneck. We will propose new types of finite-dimensional spaces for the time discretization. They allow variable time-stepping, variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially. The reason is that the basis functions of these spaces are globally smooth and compactly supported. In order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere. In this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiments.     

A parallel version of the preconditioned conjugate gradient method for boundary element equations

The parallel version of precondition techniques is developed for matrices arising from the Galerkin boundary element method for two-dimensional domains with Dirichlet boundary conditions. Results were obtained for implementations on a transputer network as well as on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited for a MIMD computer. A comparison of numerical results for iterative and direct solution methods is presented and underlines the superiority of iterative methods for large systems.