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T. Rabczuk - One of the best experts on this subject based on the ideXlab platform.
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Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase field fracture by Nonlocal Operator method.
arXiv: Numerical Analysis, 2021Co-Authors: Huilong Ren, Xiaoying Zhuang, Erkan Oterkus, Hehua Zhu, T. RabczukAbstract:The derivation of Nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on Nonlocal Operator method for the derivation of Nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase field fracture method. The Nonlocal governing equations are expressed as integral form on support and dual-support. The first example shows that the Nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding Nonlocal forms. In addition, a criterion based on the instability of the Nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate Nonlocal elasticity and the Nonlocal thin plate .
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A Nonlocal Operator method for finite deformation higher-order gradient elasticity
2021Co-Authors: Huilong Ren, Xiaoying Zhuang, Nguyen-thoi Trung, T. RabczukAbstract:We present a general finite deformation higher-order gradient elasticity theory. The governing equations of the higher-order gradient solid along with boundary conditions of various orders are derived from a variational principle using integration by parts on the surface. The objectivity of the energy functional is achieved by carefully selecting the invariants under rigid-body transformation. The third-order gradient solid theory includes more than 10.000 material parameters. However, under certain simplifications, the material parameters can be greatly reduced; down to 3. With this simplified formulation, we develop a Nonlocal Operator method and apply it to several numerical examples. The numerical analysis shows that the high gradient solid theory exhibits a stiffer response compared to a 'conventional' hyperelastic solid. The numerical tests also demonstrate the capability of the Nonlocal Operator method in solving higher-order physical problems.
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Nonlocal Operator method for the Cahn-Hilliard phase field model
Communications in Nonlinear Science and Numerical Simulation, 2021Co-Authors: Huilong Ren, Xiaoying Zhuang, Nguyen-thoi Trung, T. RabczukAbstract:Abstract In this paper we propose a Nonlocal Operator Method (NOM) for the solution of the Cahn-Hilliard (CH) equation exploiting the higher order continuity of the NOM. The method is derived based on the method of weighted residuals and implemented in 2D and 3D. Periodic boundary conditions and solid-wall boundary conditions are considered. For these boundary conditions, the highest order in the NOM scheme is 2 and 3, respectively. The proposed NOM makes use of variable support domains allowing for adaptive refinement. The generalized α -method is employed for time integration and the Newton-Raphson method to iterate nonlinearity. The performance of the proposed method is demonstrated for several two and three dimensional benchmark problems. We also implemented a CH equation with 6th order partial differential derivative and studied the influence of higher order coefficients on the pattern evolution of the phase field.
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Nonlocal Operator method with numerical integration for gradient solid
Computers & Structures, 2020Co-Authors: Xiaoying Zhuang, T. RabczukAbstract:Abstract The Nonlocal Operator method (NOM) is initially proposed as a particle-based method, which has difficulties in imposing accurately the boundary conditions of various orders. In this paper, we converted the particle-based NOM into a scheme with approximation property. The new scheme describes partial derivatives of various orders at a point by the nodes in the support and takes advantage of the background mesh for numerical integration. The boundary conditions are enforced via the modified variational principle. The particle-based NOM can be viewed as a special case of NOM with approximation property when nodal integration is used. The scheme based on numerical integration greatly improves the stability of the method. As a consequence, the requirement of the Operator energy functional in particle-based NOM is avoided. We demonstrate the capabilities of the proposed method by solving gradient elasticity problems and comparing the numerical results with exact solutions.
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a Nonlocal Operator method for solving partial differential equations
Computer Methods in Applied Mechanics and Engineering, 2020Co-Authors: Huilong Ren, Xiaoying Zhuang, T. RabczukAbstract:Abstract A Nonlocal Operator method is proposed which is generally applicable for solving partial differential equations (PDEs) of mechanical problems. The Nonlocal Operator can be regarded as the integral form “equivalent” to the differential form in the sense of a Nonlocal interaction model for solving the unknown field. The variation of a Nonlocal Operator plays an equivalent role as the derivatives of the shape functions in the meshless methods or those of the finite element method, thus it circumvents many difficulties in the calculation of shape functions and their derivatives. The Nonlocal Operator method can consistently applied with common procedure leading to the weak forms, i.e. the variational principle and the weighted residual method. Based on these, the residual and the tangent stiffness matrix can be obtained with ease. The Nonlocal Operator method is enhanced here also with an Operator energy functional to satisfy the linear consistency of the field. Higher order Nonlocal Operators and higher order Operator energy functional are hereby generalized. A highlighted of the present method is the functional derived based on the Nonlocal Operator can convert the construction of residual and stiffness matrix into a series of matrix multiplications using the predefined Nonlocal Operators. The Nonlocal strong forms of different functionals can be obtained easily via the concept of support and dual-support, the two basic elements introduced in the paper. Several numerical examples of different types of PDEs are presented in the end to show the effectiveness of the present method and also serve for validation.
Xiaoying Zhuang - One of the best experts on this subject based on the ideXlab platform.
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Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase field fracture by Nonlocal Operator method.
arXiv: Numerical Analysis, 2021Co-Authors: Huilong Ren, Xiaoying Zhuang, Erkan Oterkus, Hehua Zhu, T. RabczukAbstract:The derivation of Nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on Nonlocal Operator method for the derivation of Nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase field fracture method. The Nonlocal governing equations are expressed as integral form on support and dual-support. The first example shows that the Nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding Nonlocal forms. In addition, a criterion based on the instability of the Nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate Nonlocal elasticity and the Nonlocal thin plate .
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A Nonlocal Operator method for finite deformation higher-order gradient elasticity
2021Co-Authors: Huilong Ren, Xiaoying Zhuang, Nguyen-thoi Trung, T. RabczukAbstract:We present a general finite deformation higher-order gradient elasticity theory. The governing equations of the higher-order gradient solid along with boundary conditions of various orders are derived from a variational principle using integration by parts on the surface. The objectivity of the energy functional is achieved by carefully selecting the invariants under rigid-body transformation. The third-order gradient solid theory includes more than 10.000 material parameters. However, under certain simplifications, the material parameters can be greatly reduced; down to 3. With this simplified formulation, we develop a Nonlocal Operator method and apply it to several numerical examples. The numerical analysis shows that the high gradient solid theory exhibits a stiffer response compared to a 'conventional' hyperelastic solid. The numerical tests also demonstrate the capability of the Nonlocal Operator method in solving higher-order physical problems.
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Nonlocal Operator method for the Cahn-Hilliard phase field model
Communications in Nonlinear Science and Numerical Simulation, 2021Co-Authors: Huilong Ren, Xiaoying Zhuang, Nguyen-thoi Trung, T. RabczukAbstract:Abstract In this paper we propose a Nonlocal Operator Method (NOM) for the solution of the Cahn-Hilliard (CH) equation exploiting the higher order continuity of the NOM. The method is derived based on the method of weighted residuals and implemented in 2D and 3D. Periodic boundary conditions and solid-wall boundary conditions are considered. For these boundary conditions, the highest order in the NOM scheme is 2 and 3, respectively. The proposed NOM makes use of variable support domains allowing for adaptive refinement. The generalized α -method is employed for time integration and the Newton-Raphson method to iterate nonlinearity. The performance of the proposed method is demonstrated for several two and three dimensional benchmark problems. We also implemented a CH equation with 6th order partial differential derivative and studied the influence of higher order coefficients on the pattern evolution of the phase field.
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Nonlocal Operator method with numerical integration for gradient solid
Computers & Structures, 2020Co-Authors: Xiaoying Zhuang, T. RabczukAbstract:Abstract The Nonlocal Operator method (NOM) is initially proposed as a particle-based method, which has difficulties in imposing accurately the boundary conditions of various orders. In this paper, we converted the particle-based NOM into a scheme with approximation property. The new scheme describes partial derivatives of various orders at a point by the nodes in the support and takes advantage of the background mesh for numerical integration. The boundary conditions are enforced via the modified variational principle. The particle-based NOM can be viewed as a special case of NOM with approximation property when nodal integration is used. The scheme based on numerical integration greatly improves the stability of the method. As a consequence, the requirement of the Operator energy functional in particle-based NOM is avoided. We demonstrate the capabilities of the proposed method by solving gradient elasticity problems and comparing the numerical results with exact solutions.
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a Nonlocal Operator method for solving partial differential equations
Computer Methods in Applied Mechanics and Engineering, 2020Co-Authors: Huilong Ren, Xiaoying Zhuang, T. RabczukAbstract:Abstract A Nonlocal Operator method is proposed which is generally applicable for solving partial differential equations (PDEs) of mechanical problems. The Nonlocal Operator can be regarded as the integral form “equivalent” to the differential form in the sense of a Nonlocal interaction model for solving the unknown field. The variation of a Nonlocal Operator plays an equivalent role as the derivatives of the shape functions in the meshless methods or those of the finite element method, thus it circumvents many difficulties in the calculation of shape functions and their derivatives. The Nonlocal Operator method can consistently applied with common procedure leading to the weak forms, i.e. the variational principle and the weighted residual method. Based on these, the residual and the tangent stiffness matrix can be obtained with ease. The Nonlocal Operator method is enhanced here also with an Operator energy functional to satisfy the linear consistency of the field. Higher order Nonlocal Operators and higher order Operator energy functional are hereby generalized. A highlighted of the present method is the functional derived based on the Nonlocal Operator can convert the construction of residual and stiffness matrix into a series of matrix multiplications using the predefined Nonlocal Operators. The Nonlocal strong forms of different functionals can be obtained easily via the concept of support and dual-support, the two basic elements introduced in the paper. Several numerical examples of different types of PDEs are presented in the end to show the effectiveness of the present method and also serve for validation.
Huilong Ren - One of the best experts on this subject based on the ideXlab platform.
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Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase field fracture by Nonlocal Operator method.
arXiv: Numerical Analysis, 2021Co-Authors: Huilong Ren, Xiaoying Zhuang, Erkan Oterkus, Hehua Zhu, T. RabczukAbstract:The derivation of Nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on Nonlocal Operator method for the derivation of Nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase field fracture method. The Nonlocal governing equations are expressed as integral form on support and dual-support. The first example shows that the Nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding Nonlocal forms. In addition, a criterion based on the instability of the Nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate Nonlocal elasticity and the Nonlocal thin plate .
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A Nonlocal Operator method for finite deformation higher-order gradient elasticity
2021Co-Authors: Huilong Ren, Xiaoying Zhuang, Nguyen-thoi Trung, T. RabczukAbstract:We present a general finite deformation higher-order gradient elasticity theory. The governing equations of the higher-order gradient solid along with boundary conditions of various orders are derived from a variational principle using integration by parts on the surface. The objectivity of the energy functional is achieved by carefully selecting the invariants under rigid-body transformation. The third-order gradient solid theory includes more than 10.000 material parameters. However, under certain simplifications, the material parameters can be greatly reduced; down to 3. With this simplified formulation, we develop a Nonlocal Operator method and apply it to several numerical examples. The numerical analysis shows that the high gradient solid theory exhibits a stiffer response compared to a 'conventional' hyperelastic solid. The numerical tests also demonstrate the capability of the Nonlocal Operator method in solving higher-order physical problems.
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Nonlocal Operator method for the Cahn-Hilliard phase field model
Communications in Nonlinear Science and Numerical Simulation, 2021Co-Authors: Huilong Ren, Xiaoying Zhuang, Nguyen-thoi Trung, T. RabczukAbstract:Abstract In this paper we propose a Nonlocal Operator Method (NOM) for the solution of the Cahn-Hilliard (CH) equation exploiting the higher order continuity of the NOM. The method is derived based on the method of weighted residuals and implemented in 2D and 3D. Periodic boundary conditions and solid-wall boundary conditions are considered. For these boundary conditions, the highest order in the NOM scheme is 2 and 3, respectively. The proposed NOM makes use of variable support domains allowing for adaptive refinement. The generalized α -method is employed for time integration and the Newton-Raphson method to iterate nonlinearity. The performance of the proposed method is demonstrated for several two and three dimensional benchmark problems. We also implemented a CH equation with 6th order partial differential derivative and studied the influence of higher order coefficients on the pattern evolution of the phase field.
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a Nonlocal Operator method for solving partial differential equations
Computer Methods in Applied Mechanics and Engineering, 2020Co-Authors: Huilong Ren, Xiaoying Zhuang, T. RabczukAbstract:Abstract A Nonlocal Operator method is proposed which is generally applicable for solving partial differential equations (PDEs) of mechanical problems. The Nonlocal Operator can be regarded as the integral form “equivalent” to the differential form in the sense of a Nonlocal interaction model for solving the unknown field. The variation of a Nonlocal Operator plays an equivalent role as the derivatives of the shape functions in the meshless methods or those of the finite element method, thus it circumvents many difficulties in the calculation of shape functions and their derivatives. The Nonlocal Operator method can consistently applied with common procedure leading to the weak forms, i.e. the variational principle and the weighted residual method. Based on these, the residual and the tangent stiffness matrix can be obtained with ease. The Nonlocal Operator method is enhanced here also with an Operator energy functional to satisfy the linear consistency of the field. Higher order Nonlocal Operators and higher order Operator energy functional are hereby generalized. A highlighted of the present method is the functional derived based on the Nonlocal Operator can convert the construction of residual and stiffness matrix into a series of matrix multiplications using the predefined Nonlocal Operators. The Nonlocal strong forms of different functionals can be obtained easily via the concept of support and dual-support, the two basic elements introduced in the paper. Several numerical examples of different types of PDEs are presented in the end to show the effectiveness of the present method and also serve for validation.
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A higher order Nonlocal Operator method for solving partial differential equations
Computer Methods in Applied Mechanics and Engineering, 2020Co-Authors: Huilong Ren, Xiaoying Zhuang, T. RabczukAbstract:Abstract A higher order Nonlocal Operator method for the solution of boundary value problems is developed. The proposed higher order Nonlocal Operator brings several advantages as compared to the original Nonlocal Operator method (Ren et al., 2020) which only ensures first-order convergence. Furthermore, it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions. Only the functionals based on the Nonlocal Operators (termed as Operator functional) are needed to obtain the final discrete system of equations, which significantly facilitates the implementation. Several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order Nonlocal Operator method including the solution of the Poisson equation in 2–5 dimensional space, Kirchhoff and von Karman plate problems, incompressible elastic materials as well as phase field modeling of fracture.
Jerome Coville - One of the best experts on this subject based on the ideXlab platform.
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Singular measure as principal eigenfunction of some Nonlocal Operators
Applied Mathematics Letters, 2013Co-Authors: Jerome CovilleAbstract:In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some Nonlocal Operator. That is, we look for the existence of some particular solution (lambda, phi) of a Nonlocal Operator: integral(Omega) K(x, y)phi(y) dy + a(x)phi(x) = -lambda phi(x), where Omega subset of R-n is a bounded domain, K is a nonnegative kernel and a is continuous. We prove that for the generalised principal eigenvalue lambda(p) := sup{lambda is an element of R vertical bar there exists phi is an element of C(Omega), phi > 0 so that L-Omega[phi]+a(x)phi + lambda phi
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SINGULAR MEASURE AS PRINCIPAL EIGENFUNCTION OF SOME Nonlocal OperatorS
Applied Mathematics Letters, 2013Co-Authors: Jerome CovilleAbstract:In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some Nonlocal Operator. That is, we look for the existence of some particular solution $(\lambda,\phi)$ of a Nonlocal Operator. $$\int_{\O}K(x,y)\phi(y)\, dy +a(x)\phi(x) =-\lambda \phi(x),$$ where $\O\subset\R^n$ is an open bounded connected set, $K$ a nonnegative kernel and $a$ is continuous. We prove that for the generalised principal eigenvalue $\lambda_p:=\sup \{\lambda \in \R \, |\, \exists \, \phi \in C(\O), \phi > 0 \;\text{ so that }\; \oplb{\phi}{\O}+ a(x)\phi + \lambda\phi\le 0\}$ there exists always a solution $(\mu, \lambda_p)$ of the problem in the space of signed measure. Moreover $\mu$ a positive measure. When $\mu$ is absolutely continuous with respect to the Lebesgue measure, $\mu =\phi_p(x)$ is called the principal eigenfunction associated to $\lambda_p$. In some simple cases, we exhibit some explicit singular measures that are solutions of the spectral problem.
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SINGULAR MEASURE AS PRINCIPAL EIGENFUNCTION OF SOME Nonlocal OperatorS
Applied Mathematics Letters, 2013Co-Authors: Jerome CovilleAbstract:Abstract In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some Nonlocal Operator. That is, we look for the existence of some particular solution ( λ , ϕ ) of a Nonlocal Operator: ∫ Ω K ( x , y ) ϕ ( y ) d y + a ( x ) ϕ ( x ) = − λ ϕ ( x ) , where Ω ⊂ R n is a bounded domain, K is a nonnegative kernel and a is continuous. We prove that for the generalised principal eigenvalue λ p ≔ sup { λ ∈ R ∣ ∃ ϕ ∈ C ( Ω ) , ϕ > 0 so that L Ω [ ϕ ] + a ( x ) ϕ + λ ϕ ≤ 0 } there exists always a solution ( d μ , λ p ) of the problem in the space of positive measure. When d μ is absolutely continuous with respect to the Lebesgue measure, d μ = ϕ p ( x ) d x is called the principal eigenfunction associated with λ p . In some simple cases, we exhibit some explicit singular measures that are solutions of the spectral problem.
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Remarks on the strong maximum principle for Nonlocal Operators
Electronic Journal of Differential Equations, 2008Co-Authors: Jerome CovilleAbstract:In this note, we study the existence of a strong maximum principle for the Nonlocal Operator M(x) := integral(G) J(g)u(x * g(-1))d mu(g) - u(x), where G is a topological group acting continuously on a Hausdorff space X and u is an element of C(X). First we investigate the general situation and derive a pre-maximum principle. Then we restrict our analysis to the case of homogeneous spaces (i.e., X = G/H). For such Hausdorff spaces, depending on the topology, we give a condition on J such that a strong maximum principle holds for M. We also revisit the classical case of the convolution Operator (i.e. G - (R-n, +), X = R-n, d mu = dy)
Ki-ahm Lee - One of the best experts on this subject based on the ideXlab platform.
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Boundary regularity for Nonlocal Operators with kernels of variable orders
Journal of Functional Analysis, 2019Co-Authors: Minhyun Kim, Panki Kim, Jaehun Lee, Ki-ahm LeeAbstract:Abstract We study the boundary regularity of solutions of the Dirichlet problem for the Nonlocal Operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized Holder space. We prove that there exists a unique viscosity solution of L u = f in D, u = 0 in R n ∖ D , where D is a bounded C 1 , 1 open set, and that the solution u satisfies u ∈ C V ( D ) and u / V ( d D ) ∈ C α ( D ) with the uniform estimates, where V is the renewal function and d D ( x ) = dist ( x , ∂ D ) .
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Boundary regularity for Nonlocal Operators with kernels of variable orders.
arXiv: Analysis of PDEs, 2018Co-Authors: Minhyun Kim, Panki Kim, Jaehun Lee, Ki-ahm LeeAbstract:We study the boundary regularity of solutions of the Dirichlet problem for the Nonlocal Operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized H\"older space. We prove that there exists a unique viscosity solution of $Lu = f$ in $D$, $u=0$ in $\mathbb{R}^n \setminus D$, where $D$ is a bounded $C^{1,1}$ open set, and that the solution $u$ satisfies $u \in C^V(D)$ and $u/V(d_D) \in C^\alpha (D)$ with the uniform estimates, where $V$ is the renewal function and $d_D(x) = \mbox{dist}(x, \partial D)$.