Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

The weighted gradient reproducing kernel collocation method is introduced to recover the heat source described by Poisson’s equation. As it is commonly known that there is no unique solution to the inverse heat source problem, the weak solution based on a priori assumptions is considered herein. In view of the fourth-order partial differential equation (PDE) in the mathematical model, the high-order gradient reproducing kernel approximation is introduced to efficiently untangle the problem without calculating the high-order derivatives of reproducing kernel shape functions. The weights of the weighted collocation method for high-order inverse analysis are first determined. In the benchmark analysis, the unclear illustration in the literature is clarified, and the correct interpretation of numerical results is given particularly. Two mathematical formulations with four examples are provided to demonstrate the viability of the method, including the extreme cases of the limited accessible boundary.

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Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method

Mathematics ◽

10.3390/math8081297 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1297 ◽

Cited By ~ 1

Author(s):

Judy P. Yang ◽

Hon Fung Samuel Lam

Keyword(s):

Collocation Method ◽

Reproducing Kernel ◽

Outer Boundary ◽

Computational Cost ◽

Weak Form ◽

High Order ◽

Unknown Boundary ◽

Shape Functions ◽

High Order Derivative ◽

Special Cases

The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically increase, especially when dealing with strong-from equations where high-order derivative operators are required as compared to weak-form approaches for obtaining results with promising levels of accuracy. Under the framework of gradient approximation, the derivatives of reproducing kernel shape functions can be constructed synchronically, thereby alleviating the complexity in computation. In view of this, the present work first introduces the weighted high-order gradient reproducing kernel collocation method in the inverse analysis. The convergence of the method is examined through the weights imposed on the boundary conditions. Then, several configurations of multiply connected domains are provided to numerically investigate the stability and efficiency of the method. To reach the desired accuracy in detecting the outer boundary for two special cases, special treatments including allocation of points and use of ghost points are adopted as the solution strategy. From four benchmark examples, the efficacy of the method in detecting the unknown boundary is demonstrated.

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A gradient reproducing kernel collocation method for high order differential equations

Computational Mechanics ◽

10.1007/s00466-019-01724-0 ◽

2019 ◽

Vol 64 (5) ◽

pp. 1421-1454 ◽

Cited By ~ 7

Author(s):

Ashkan Mahdavi ◽

Sheng-Wei Chi ◽

Huiqing Zhu

Keyword(s):

Differential Equations ◽

Collocation Method ◽

Reproducing Kernel ◽

High Order

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Reproducing kernel mesh-free collocation analysis of structural vibrations

Engineering Computations ◽

10.1108/ec-10-2018-0439 ◽

2019 ◽

Vol 36 (3) ◽

pp. 734-764 ◽

Cited By ~ 6

Author(s):

Dongliang Qi ◽

Dongdong Wang ◽

Like Deng ◽

Xiaolan Xu ◽

Cheng-Tang Wu

Keyword(s):

Collocation Method ◽

Fourth Order ◽

Reproducing Kernel ◽

Basis Functions ◽

Structural Vibration ◽

Structural Vibrations ◽

Content Type ◽

Mesh Free ◽

Odd Degree ◽

Vibration Problems

PurposeAlthough high-order smooth reproducing kernel mesh-free approximation enables the analysis of structural vibrations in an efficient collocation formulation, there is still a lack of systematic theoretical accuracy assessment for such approach. The purpose of this paper is to present a detailed accuracy analysis for the reproducing kernel mesh-free collocation method regarding structural vibrations.Design/methodology/approachBoth second-order problems such as one-dimensional (1D) rod and two-dimensional (2D) membrane and fourth-order problems such as Euler–Bernoulli beam and Kirchhoff plate are considered. Staring from a generic equation of motion deduced from the reproducing kernel mesh-free collocation method, a frequency error measure is rationally attained through properly introducing the consistency conditions of reproducing kernel mesh-free shape functions.FindingsThis paper reveals that for the second-order structural vibration problems, the frequency accuracy orders arepand (p− 1) for even and odd degree basis functions; for the fourth-order structural vibration problems, the frequency accuracy orders are (p− 2) and (p− 3) for even and odd degree basis functions, respectively, wherepdenotes the degree of the basis function used in mesh-free approximation.Originality/valueA frequency accuracy estimation is achieved for the reproducing kernel mesh-free collocation analysis of structural vibrations, which can effectively underpin the practical applications of this method.

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A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type

Advances in Mathematical Physics ◽

10.1155/2013/787891 ◽

2013 ◽

Vol 2013 ◽

pp. 1-14 ◽

Cited By ~ 2

Author(s):

Yang Liu ◽

Hong Li ◽

Zhichao Fang ◽

Siriguleng He ◽

Jinfeng Wang

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Finite Element ◽

Error Estimates ◽

Fourth Order ◽

A Priori ◽

Parabolic Type ◽

A Priori Error Estimates ◽

Order Partial Differential Equation ◽

Priori Error Estimates

We propose and analyze a new numerical method, called a coupling method based on a new expanded mixed finite element (EMFE) and finite element (FE), for fourth-order partial differential equation of parabolic type. We first reduce the fourth-order parabolic equation to a coupled system of second-order equations and then solve a second-order equation by FE method and approximate the other one by a new EMFE method. We find that the new EMFE method’s gradient belongs to the simple square integrable(L2(Ω))2space, which avoids the use of the classicalH(div; Ω) space and reduces the regularity requirement on the gradient solutionλ=∇u. For a priori error estimates based on both semidiscrete and fully discrete schemes, we introduce a new expanded mixed projection and some important lemmas. We derive the optimal a priori error estimates inL2andH1-norm for both the scalar unknownuand the diffusion termγand a priori error estimates in(L2)2-norm for its gradientλand its fluxσ(the coefficients times the negative gradient). Finally, we provide some numerical results to illustrate the efficiency of our method.

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