An interface integral equation method for solving general multi-medium mechanics problems

Numerical Meth Engineering

2023

In this article, a novel weak‐form zonal Petrov–Galerkin free element method is proposed for two‐ and three‐dimensional linear mechanical problems. By absorbing the advantages of finite block method and strong‐form finite element method, the block mapping technique is used in the free element method. Combining the characteristics of the meshless local Petrov–Galerkin method, the local Petrov–Galerkin formulation based on the zonal free element method is formed at last. Besides, the local integral domain selected in the local collocation element is circular or spherical to simplify programming. The transformation of the local integral domain between the physical and normalized spaces is given for two‐ and three‐dimensional problems. The comparison of accuracy and convergence between the new proposed Petrov–Galerkin method and the conventional methods is carried out. Some challenging examples including fracture mechanics problems and a complex 3D problem are given to validate the convergence and accuracy of the proposed method.

Section: Introductionmentioning

confidence: 99%

Petrov–Galerkin zonal free element method for 2D and 3D mechanical problems

Numerical Meth Engineering

2023

“…BEM only needs to discretize the boundary of the problem into elements, which not only can reduce the dimension, but also can easily simulate some special cases like crack and stress concentration [8], [12]. However, the final system of BEM has a dense coefficient matrix and the fundamental solutions are derived from the linear problems, which severely limits the application of BEM in large problems, nonlinear and nonhomogeneous problems [12], [15]. Different from mesh-dependent methods, MFM only needs a group of distributed nodes in the computational domain and barely needs elements.…”

Section: Introductionmentioning

confidence: 99%

Bem–edm Coupled Analysis of Multi-Scale Problems

Zheng

Boundary Elements and Other Mesh Reduction Methods XLII

Peng

2019

Self Cite

In this paper, the element differential method (EDM), a new numerical method proposed recently, is coupled with the boundary element method (BEM), a traditional numerical method, for solving general multi-scale heat conduction problems. The basic algebraic equations in BEM are formulated in terms of temperatures and heat fluxes, which are the same as those in EDM. So, when coupling these two methods, we do not need to transform the variables like the thermal loads into heat fluxes as done with the finite element method (FEM). The key task in the proposed coupled method is to use the temperature consistency condition and the flux equilibrium equation at interface nodes of the two methods to eliminate all BEM nodes except for those on the interfaces. The detailed elimination process is presented in this paper, which can result in the final system of equations without iteration. The coefficient matrix of the final coupled system is sparse, even though a small part is dense. The coupled method inherits the advantage of EDM in flexibility and computational efficiency and the advantage of BEM in the robustness of treating multi-scale problems. A numerical example is given to demonstrate the correctness of this coupled method.

“…In the weak‐form technique, the frequently used methods are the finite element method (FEM), boundary element method (BEM), finite volume method (FVM), and a part of mesh free methods (MFM) . In this technique, the spatially partial derivatives of physical variables appearing in the governing PDEs are converted into some terms, including the physical variables and their lower orders of derivatives based on a mathematical principle, eg, the variational principle, or a mechanical principle, eg, the energy principle .…”

Section: Introductionmentioning

confidence: 99%

“…The drawbacks of FEM are mainly embodied in that a variational or a virtual work principle is needed to establish the FEM algorithm and the resulted integrals need to be evaluated over the discretized elements. Comparing to FEM, BEM only needs to discretize the boundary of the problem into elements, which not only can reduce the dimensions of the problem but also can easily simulate the stress concentration behaviors . One of the drawbacks of BEM is that the fundamental solutions are derived from linear problems, and therefore it is difficult to establish a pure BEM algorithm for nonlinear and nonhomogeneous problems .…”

Section: Introductionmentioning

confidence: 99%

“…Comparing to FEM, BEM only needs to discretize the boundary of the problem into elements, which not only can reduce the dimensions of the problem but also can easily simulate the stress concentration behaviors . One of the drawbacks of BEM is that the fundamental solutions are derived from linear problems, and therefore it is difficult to establish a pure BEM algorithm for nonlinear and nonhomogeneous problems . Different from these mesh‐dependent methods, MFM only needs a group of distributed points in the computational domain.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Element differential method and its application in thermal‐mechanical problems

Numerical Meth Engineering

Yang

et al. 2017

Self Cite

SummaryIn this paper, a new numerical method, element differential method (EDM), is proposed for solving general thermal-mechanical problems. The key point of the method is the direct differentiation of the shape functions of Lagrange isoparametric elements used to characterize the geometry and physical variables. A set of analytical expressions for computing the first-and second-order partial derivatives of the shape functions with respect to global coordinates are derived. Based on these expressions, a new collocation method is proposed for establishing the system of equations, in which the equilibrium equations are collocated at nodes inside elements, and the traction equilibrium equations are collocated at interface nodes between elements and outer surface nodes of the problem. Attributed to the use of the Lagrange elements that can guarantee the variation of physical variables consistent through all elemental nodes, EDM has higher stability than the traditional collocation method. The other main features of EDM are that no mathematical or mechanical principles are required to set up the system of equations and no integrals are involved to form the coefficients of the system. A number of numerical examples of 2-and 3-dimensional problems are given to demonstrate the correctness and efficiency of the proposed method. | INTRODUCTIONIn mathematics, the thermal-mechanical problem is one of the boundary value problems of the second-order partial differential equations (PDEs). The key task to solve this type of PDEs is to treat the involved second-order partial derivatives of related physical variables with respect to the global coordinates. A number of successful numerical methods are available to do this, which can be classified into 2 types: the weak-form technique and the strong-form technique.