Element differential method and its application in thermal‐mechanical problems

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.

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“…After solving the system, we can use eqns (25) and (24) to calculate T . Thus, all the unknowns of computational domain can be obtained.…”

Section: Boundary Element Methodsmentioning

confidence: 99%

“…Recently, a new numerical method called element differential method (EDM) is proposed by Gao et al [23]- [25] and Cui et al [26]. It is also a flexible method like FEM and can be used in most engineering problems.…”

Section: Introductionmentioning

confidence: 99%

Bem–edm Coupled Analysis of Multi-Scale Problems

Zheng

Gao

Peng

2019

Boundary Elements and Other Mesh Reduction Methods XLII

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“…The formula of shape functions and its first or second order can be obtained in the reference papers [11,12].…”

Section: Using Edm To Discretize the Governing Equationsmentioning

confidence: 99%

“…In recent years, Gao et al [11][12][13][14][15][16][17][18][19][20] proposed a new strong-form numerical method called element differential method (EDM). Like traditional FEM, EDM also uses isoparametric elements to discretize the computational domain and approximate physical variables.…”

Section: Introductionmentioning

confidence: 99%

Multi-physics coupling analysis of rope-sealed structures with braided ceramic fibres by element differential method

Zheng¹,

Gao²,

Liu³

et al. 2021

Int. J. CMEM

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With the rapid development of hypersonic vehicles in recent years, high-temperature seal technology has become more and more essential. Recently, a rope-sealed structure with braided ceramic fibres has been designed for hypersonic vehicles. The ceramic fibres in the structure have the characteristics of high temperature strength, so that they make the sealed structure suitable for working under a high temperature. Meanwhile, when subjected to an external force, braided fibres can produce a buffer force at the ceramic interface, so that it can maintain the good performance of the whole sealed structure. But up to now, only a few researches have been conducted on this kind of structures. In this paper, a simplified thermal-mechanical seepage coupling model is proposed to simulate the complicated physical process for this kind of structures. Meanwhile, a new numerical method called element differential method (EDM) is used to calculate the coupling problem because it has great advantages in solving multiphysics coupling problems. What is more, some experiments are used to obtain the leakages when the sealed structure is under service. And finally, by referring the experimental results, the authors establish a series of material parameter relationships for the sealed structure and also verify the reasonability of the proposed multi-physics coupling model.

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“…Then, we transform the irregular element into a regular domain as done in FEM. The variables and Cartesian coordinates can be obtained by interpolation

\overset{true‾}{ϕ} = \sum N_{i} false(boldξ false) ϕ^{i}, x = \sum N_{i} false(boldξ false) {boldx}^{i},

where

x = false(x_{1} x_{2} \dots x_{N_{D}} false)

and

ξ = false(ξ_{1} ξ_{2} \dots ξ_{N_{D}} false)

represent the Cartesian coordinates and Natural coordinates, respectively, N i represents interpolation functions or called the shape functions in FEM which can be found in References 36 in detail, and N D represent the spatial dimension of the problem. For 3D problem, ( x 1 , x 2 , x 3 ) = ( x , y , z ) and ( ξ 1 , ξ 2 , ξ 3 ) = ( ξ , η , ζ ).…”

Section: Momentum Equationmentioning

confidence: 99%

A free element scheme for simulating two‐ and three‐dimensional incompressible fluid flow

Liu

Gao

2020

Numerical Methods in Fluids

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In this work, Free Element method (FECM) is extended to solve incompressible Navier‐Stokes equations. The momentum equations are discretized by FECM, which permits overlapped elements. Then, the velocities at midpoints are interpolated by improved Momentum Interpolation Method to avoid oscillation caused by decoupling of velocity and pressure. At last, the pressure correction equation is solved to make sure the results satisfying continuity equation. The interpolation of midpoint's velocity is proved to be independent of the under‐relaxation factor and time step size. The deferred correction is employed for utilization of higher‐order discretization of convective terms. Taylor‐Green vortex, flow over a cylinder, lid‐driven flow, cooling tunnel, and flow over airfoil are simulated. The results computed by the proposed method are compared with exact solutions and benchmark solutions, which indicates that the new method has the second order accuracy in space. What is more, the proposed method works well on random node distributions as well, which means that the method is much robust.

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Element differential method and its application in thermal‐mechanical problems

Cited by 47 publications

References 44 publications

Bem–edm Coupled Analysis of Multi-Scale Problems

Bem–edm Coupled Analysis of Multi-Scale Problems

Multi-physics coupling analysis of rope-sealed structures with braided ceramic fibres by element differential method

A free element scheme for simulating two‐ and three‐dimensional incompressible fluid flow

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