Reproducing Kernel Approximation Research Articles

A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

A gradient reproducing kernel based stabilized collocation method (GRKSCM) to numerically solve complicated nonlinear Korteweg-de Vries (KdV) equation is proposed in this paper. The acquisition of GRK through high-order consistency conditions reduces the complexity of RK derivative operations and remarkably improves the effectiveness of the proposed method by directly forming GRK approximations. Owing to the fulfillment of high-order integration constraints, the SCM achieves exact integration in the domain and on boundaries. Von Neumann analysis is utilized to establish the stability criteria for GRKSCM when combined with forward difference temporal discretization. The effectiveness of the GRKSCM method in solving the KdV equation is investigated through six numerical examples. The examples include the motion of the single solitary wave propagation, interaction between two solitary waves and interaction among three solitary waves. Furthermore, the behavior of a two-dimensional solitary wave and the propagation of a three-dimensional solitary wave are also numerically investigated. The numerical outcomes confirm that GRKSCM provides high accuracy in comparison with analytical solutions. In addition, the invariants and error analysis show the conservation of our proposed GRKSCM method.

A neural network-based enrichment of reproducing kernel approximation for modeling brittle fracture

Numerical modeling of localizations is a challenging task due to the evolving rough solution in which the localization paths are not predefined. Despite decades of efforts, there is a need for innovative discretization-independent computational methods to predict the evolution of localizations. In this work, an improved version of the neural network-enhanced Reproducing Kernel Particle Method (NN-RKPM) is proposed for modeling brittle fracture. In the proposed method, a background reproducing kernel (RK) approximation defined on a coarse and uniform discretization is enriched by a neural network (NN) approximation under a Partition of Unity framework. In the NN approximation, the deep neural network automatically locates and inserts regularized discontinuities in the function space. The NN-based enrichment functions are then patched together with RK approximation functions using RK as a Partition of Unity patching function. The optimum NN parameters defining the location, orientation, and displacement distribution across location together with RK approximation coefficients are obtained via the energy-based loss function minimization. To regularize the NN-RK approximation, a constraint on the spatial gradient of the parametric coordinates is imposed in the loss function. Analysis of the convergence properties shows that the solution convergence of the proposed method is guaranteed. The NN enrichment allows the modeling of evolving cracks by a fixed coarse RK discretization without adaptive refinement for enhanced computational efficiency. The effectiveness of the proposed method is demonstrated by a series of numerical examples involving damage propagation and branching.

Numerical simulation of 3D fracture propagation problem with reproducing kernel peridynamic method

The non-ordinary state-based peridynamic (NOSB-PD) method has significant advantages in solving crack propagation problems due to its non-local characteristics, but it also faces problems such as numerical oscillations and boundary effects. In order to solve the above defects, the paper first discusses the NOSB-PD method within the Galerkin framework. After that, the peridynamic differential operator (PDDO) approximation was introduced by extending the calculation of the deformation gradient tensor F in NOSB-PD. Since the zero-order approximations in the PDDO approximation and the reproducing kernel (RK) approximation are identical, the distinctions between the two approximations have been thoroughly analysed. Although the two approximations show the same consistency, only the RK approximation satisfies the integrable condition. Therefore, a RK-PD coupling method is proposed in the paper, and an implicit iterative algorithm suited to the RK-PD method is presented as well. At last, several numerical examples demonstrate that the proposed method does not suffer from numerical oscillations and can be used to solve the three-dimensional crack propagation problems.

A kernel‐based method for solving the time‐fractional diffusion equation

AbstractIn this paper, we focus on the development and study of a numerical method based on the idea of kernel‐based approximation and finite difference discretization to obtain the solution for the time‐fractional diffusion equation. Using the theory of reproducing kernel, reproducing kernel functions with a polynomial form will be established in polynomial reproducing kernel spaces spanned by the Chebychev basis polynomials. In the numerical method, first the time‐fractional derivative term in the aforementioned equation is approximated by using the finite difference scheme. Then, by the help of collocation method based on reproducing kernel approximation, we will illustrate how to derive the numerical solution in polynomial reproducing kernel space. Finally, to support the accuracy and efficiency of the numerical method, we provide several numerical examples. In numerical experiments, the quality of approximation is calculated by absolute error and discrete error norms.

Gradient reproducing kernel based Hermite collocation method (GHCM) for eigenvalue analysis of functionally graded thin plates with in-plane material

A meshfree Hermite collocation method based on gradient reproducing kernel approximations is proposed for the eigenvalue analysis of thin functionally graded plates with in-plane material inhomogeneity. Compared with direct collocation method (DCM), Hermite collocation method (HCM) can improve the accuracy of the solutions, especially on the boundaries. Moreover, HCM results in a determined system for the eigenproblems while DCM leads to an over-determined system where the solutions may not be the real results. Since derivative calculations of the reproducing kernel function are complex and time-consuming, gradient reproducing kernel approximations are introduced for solving the thin plate problems to avoid the high order direct differentiations which are required in the thin plate strong form solutions. Proper weights that should be imposed for the boundary conditions to balance the errors in the solutions are derived for both DCM and HCM. Convergence studies are also derived to evaluate the convergence of these two methods. Comparison studies with analytical solutions indicate the high accuracy and good performance of the proposed method. Numerical simulations demonstrate that material inhomogeneity has considerable effects on the natural frequencies and mode shapes.

Stabilized Lagrange Interpolation Collocation Method: A meshfree method incorporating the advantages of finite element method

Since most approximation functions in meshfree methods are rational functions which do not possess the Kronecker delta property, how to achieve exact integration and accurately impose the essential boundary conditions are two typical difficulties for meshfree methods. In this paper, a new stabilized Lagrange interpolation collocation method (SLICM) is proposed in which the Lagrange interpolation (LI) is employed for the approximation in a meshfree method. This method can satisfy the high order integration constraints which can conserves the high order consistency conditions in the integration form. This property leads to the exact integration in the subdomains and optimal convergence for the proposed method. Meanwhile, performing the integration in subdomains can also reduce the condition number of discrete matrix, which improves the stability of the algorithm. Since the Lagrange interpolation approximation has Kronecker delta property, the essential boundary conditions can be simply and exactly imposed like the finite element method, which further improves the accuracy of this method. Convergence studies present that the same convergence rate can be attained for utilizing the odd and even order LI shape functions, while the convergence rate is reduced if the odd order basis function is employed in the reproducing kernel (RK) approximation. Numerical examples validate the high accuracy and convergence as well as good stability of the presented method, which can outperform the direct collocation method and the stabilized collocation method based on RK approximation.

A highly efficient and accurate Lagrangian–Eulerian stabilized collocation method (LESCM) for the fluid–rigid body interaction problems with free surface flow

A Lagrangian–Eulerian stabilized collocation method (LESCM) for the fluid–structure interaction problems involving free surface flow is presented in this work, in which the structure is modeled by a rigid body. This method is an evolution of the material point method and particle-in-cell methods which are based on the hybrid Lagrangian–Eulerian description. The problem domain of the fluid and structures is discretized into the Lagrangian particles which carry the information, and the problem domain covering the entire movement space is discretized into the uniformly distributed Eulerian background nodes. The coupling governing equations of the fluid, structures and interfaces are solved by the meshfree stabilized collocation method (SCM) employing the reproducing kernel (RK) approximation on the Eulerian nodes. The solution is very efficient since the Eulerian nodes are set to be the initial positions in each time step and it is no need to reconstruct the shape function. The information mappings between the Lagrangian particles and the Eulerian nodes are also conducted by the RK approximation which can keep the mass and momentum conservation of the solution. The cell-cut algorithm is introduced to couple the fluid and the structures which can solve the fluid pressure and the fluid–structure interactional force simultaneously and avoid the complicated iterations of the traditional interaction algorithms. Several numerical examples including the collapse of water column with a rigid barrier, water entry of a half-buoyant circular cylinder and a rigid box rotating and sinking in water are simulated, which demonstrate the high accuracy, high efficiency and good stability of the proposed method. This method can be extensively applied to the engineering applications of fluid–rigid body interactions.

Meshfree Collocation Framework for Multi-Phase Coupling Nonlinear Dynamic System in a Porous Enclosure

This study is aimed at establishing an effective incremental-iterative framework, for the first time, using reproducing kernel collocation method for solving multi-phase dynamic coupling problems in a porous enclosure. The central difference method is utilized in the temporal discretization while direct collocation with reproducing kernel approximation is adopted in the spatial discretization, in particular, with Newton–Raphson method for nonlinear iteration. Under the proposed numerical framework, a determined and well-conditioned nonlinear system is ensured during nonlinear iteration. In addition to the traditional mesh-based weak-form methods, it is shown that the present study provides an alternative effective approach to solve unsteady-state double-diffusive convection problems.

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.

A duality‐based coupling of Cosserat crystal plasticity and phase field theories for modeling grain refinement

AbstractHigh‐rate deformation processes of metals entail intense grain refinement and special attention needs to be paid to capture the evolution of microstructure. In this article, a new formulation for coupling Cosserat crystal plasticity and phase field is developed. A common approach is to penalize kinematic incompatibility between lattice orientation and displacement‐based elastic rotation. However, this can lead to significant solution sensitivity to the penalty parameter, resulting in low accuracy and convergence rates. To address these issues, a duality‐based formulation is developed which directly imposes the rotational kinematic compatibility. A weak inf‐sup‐based skew‐symmetric stress projection is introduced to suppress instabilities present in the dual formulation. An additional least squares stabilization is introduced to suppress the spurious lattice rotation with a suitable parameter range derived analytically and validated numerically. The required high‐order continuity is attained by the reproducing kernel approximation. It is observed that equal order displacement‐rotation‐phase field approximations are stable, which allows efficient employment of the same set of shape functions for all independent variables. The proposed formulation is shown to yield superior accuracy and convergence with marginal parameter sensitivity compared to the penalty‐based approach and successfully captures the dominant rotational recrystallization mechanism including block dislocation structures and grain boundary migration.

A stabilized collocation method based on the efficient gradient reproducing kernel approximations for the boundary value problems

The meshfree strong form direct collocation method (DCM) can address the domain integration issues involved in the Galerkin meshfree methods and outperform them in the efficiency. However, the accuracy and stability of the DCM may not match the Galerkin meshfree methods in solving the complex problems. Moreover, strong form demands the high order derivatives calculation of the approximation function which increases its computational complexity and costs. Therefore, in the present study, we introduce a stabilized collocation method (SCM) associated with the gradient reproducing kernel (GRK) approximations. This method can improve the accuracy and stability of the DCM by proposing the exact numerical integration in the subdomains, and construct the GRK approximations instead of directly taking the reproducing kernel (RK) derivatives to simplify the computational complexity and promote the efficiency. The exact local integration of the SCM is much more efficient than the global integration of the Galerkin meshfree methods which makes the SCM keep the high efficiency as the DCM. Comparison analyses among the RK-CM, GRK-CM, RK-SCM and GRK-SCM are detailedly studied. Numerical simulations demonstrate that GRK-SCM can surpass the conventional RK-CM in accuracy, stability and efficiency, followed by the RK-SCM and then the GRK-CM, which provides an efficient and stable meshfree method for solving the boundary value problems.

A meshfree method with gradient smoothing for free vibration and buckling analysis of a strain gradient thin plate

In this paper, a meshfree Galerkin approach is presented for analysis of free vibration and buckling of a strain gradient thin plate. A cubic moving least square or reproducing kernel approximation with C2 continuity is used to ensure convergence. Due to smoothness of the meshfree approximation, the only variable is the deflection of a thin plate. To improve the computational accuracy, a consistent integration scheme with gradient smoothing is introduced to construct stiffness and geometrical stiffness matrices. The effectiveness of such an approach is confirmed by numerical results, and it is superior to a standard Gauss integration based meshfree Galerkin method. Further, the influence of classical and high order boundary conditions is investigated on the natural frequencies and critical buckling loads of strain gradient thin plates.

Direct Collocation with Reproducing Kernel Approximation for Two-Phase Coupling System in a Porous Enclosure

The direct strong-form collocation method with reproducing kernel approximation is introduced to efficiently and effectively solve the natural convection problem within a parallelogrammic enclosure. As the convection behavior in the fluid-saturated porous media involves phase coupling, the resulting system is highly nonlinear in nature. To this end, the local approximation is adopted in conjunction with Newton–Raphson method. Nevertheless, to unveil the performance of the method in the nonlinear analysis, only single thermal natural convection is of major concern herein. A unit square is designated as the model problem to investigate the parameters in the system related to the convergence; several benchmark problems are used to verify the accuracy of the approximation, in which the stability of the method is demonstrated by considering various initial conditions, disturbance of discretization, inclination, aspect ratio, and reproducing kernel support size. It is shown that a larger support size can be flexible in approximating highly irregular discretized problems. The derivation of explicit operators with two-phase variables in solving the nonlinear system using the direct collocation is carried out in detail.

Convergence analysis of reproducing kernel particle method to elliptic eigenvalue problem

AbstractIn this work we aim to provide a fundamental theory of the reproducing kernel particle method for solving elliptic eigenvalue problems. We concentrate on the convergence analysis of eigenvalues and eigenfunctions, as well as the optimal estimations which are shown to be related to the reproducing degree, support size, and overlapping number in the reproducing kernel approximation. The theoretical analysis has also demonstrated that the order of convergence for eigenvalues is in the square of the order of convergence for eigenfunctions. Numerical results are also presented to validate the theoretical analysis.

Asymptotically Compatible Reproducing Kernel Collocation and Meshfree Integration for Nonlocal Diffusion

Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme which is stable. In addition, there is a large computational cost for assembling the stiffness matrix of the nonlocal problem because high order Gaussian quadrature is usually needed to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares (GMLS).

An efficient meshfree gradient smoothing collocation method (GSCM) using reproducing kernel approximation

Strong-form direct collocation method based on the reproducing kernel (RK) approximation has been criticized frequently by the complexity and time-consuming computation of the derivatives, and it is rather hard to use the low-order basis functions to improve the efficiency. In this paper, we propose an efficient meshfree gradient smoothing collocation method (GSCM) based on the RK approximation which adopts the gradient smoothing technique for the calculation of the RK derivatives. Low-order basis functions such as constant functions can be utilized for the approximation in the numerical solutions of the elasticity problems which can greatly improve the efficiency. Conforming and nonconforming meshes can be constructed for the numerical integration of the gradient smoothing. Constraints of the numerical integration are established where the proper positions of the integration points and the corresponding weighs for the integration are derived to keep the consistency of the RK shape function and its smoothed gradients. These guarantee the accuracy and convergence of the proposed method. Numerical results demonstrate that the presented method can outmatch the conventional direct collocation method (DCM) in accuracy, stability and computational efficiency, and generally the GSCM-II using a double gradient smoothing performs better than the GSCM-I and the super-convergent gradient smoothing meshfree collocation method (SGSMC) using a single gradient smoothing.

A meshfree stabilized collocation method (SCM) based on reproducing kernel approximation

Meshfree direct collocation method (DCM) based on strong form suffers low accuracy and instability compared with Galerkin-based meshfree methods. Meshfree weighted least squares collocation method (WLSCM) can improve the accuracy and stability by minimizing the least squares functional corresponding to the governing equation and boundary conditions. However, usage of more discrete points in the least squares solution increases the computational cost which notably reduces the efficiency. In this paper, we propose a new meshfree stabilized collocation method (SCM) and introduce the reproducing kernel function as the approximation. Auxiliary collocation points located in the local subdomains are employed for the stabilization. Their proper positions and corresponding weighting factors for integration using constant and cubic B-spline weighting functions are derived to keep the consistency conditions. Therefore, in this method, the approximation can satisfy the consistency conditions not only on the points, but also in the subdomains. The suggested method has the comparable accuracy and stability compared with WLSCM, and the computation efficiency can match up to DCM. Numerical simulations demonstrate that SCM can surpass DCM in accuracy and stability, and outperform WLSCM in efficiency and conditioning of the matrix according to the discrete equations, where the condition number of matrix is related to the stability.

Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

In this work, we study reproducing kernel (RK) collocation method for peridynamic Navier equation. In the first part, we apply a linear RK approximation to both displacement and dilatation, and then back-substitute dilatation and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. In the second part, we apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lamé parameters satisfy λ≥μ to avoid extra assumptions on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.

A physics-constrained data-driven approach based on locally convex reconstruction for noisy database

Physics-constrained data-driven computing is an emerging hybrid approach that integrates universal physical laws with data-driven models of experimental data for scientific computing. A new data-driven simulation approach coupled with a locally convex reconstruction, termed the local convexity data-driven (LCDD) computing, is proposed to enhance accuracy and robustness against noise and outliers in data sets in the data-driven computing. In this approach, for a given state obtained by the physical simulation, the corresponding optimum experimental solution is sought by projecting the state onto the associated local convex manifold reconstructed based on the nearest experimental data. This learning process of local data structure is less sensitive to noisy data and consequently yields better accuracy. A penalty relaxation is also introduced to recast the local learning solver in the context of non-negative least squares that can be solved effectively. The reproducing kernel approximation with stabilized nodal integration is employed for the solution of the physical manifold to allow reduced stress–strain data at the discrete points for enhanced effectiveness in the LCDD learning solver. Due to the inherent manifold learning properties, LCDD performs well for high-dimensional data sets that are relatively sparse in real-world engineering applications. Numerical tests demonstrated that LCDD enhances nearly one order of accuracy compared to the standard distance-minimization data-driven scheme when dealing with noisy database, and a linear exactness is achieved when local stress–strain relation is linear.

Harmonic-Enriched Reproducing Kernel Approximation for Highly Oscillatory Differential Equations

AbstractThe harmonic-enriched reproducing kernel (HRK) approximation together with collocation method is introduced to circumvent the discretization restriction for highly oscillatory partial diffe...