Semi-Lagrangian reproducing kernel particle method for fragment-impact problems

Fragment-impact problems exhibit excessive material distortion and complex contact conditions that pose considerable challenges in mesh based numerical methods such as the finite element method (FEM). A semi-Lagrangian reproducing kernel particle method (RKPM) is proposed for fragment-impact modeling to alleviate mesh distortion difficulties associated with the Lagrangian FEM and to minimize the convective transport effect in the Eulerian or Arbitrary Lagrangian Eulerian FEM. A stabilized non-conforming nodal integration with boundary correction for the semi-Lagrangian RKPM is also proposed. Under the framework of semi-Lagrangian RKPM, a kernel contact algorithm is introduced to address multi-body contact. Stability analysis shows that temporal stability of the kernel contact algorithm is related to the velocity gradient between two contacting bodies. The performance of the proposed methods is examined by numerical simulation of penetration processes.     

The dimension splitting reproducing kernel particle method for three-dimensional potential problems

In this paper, the dimension splitting reproducing kernel particle method (DSRKPM) for three-dimensional (3D) potential problems is presented. In the DSRKPM, a 3D potential problem can be transformed into a series of two-dimensional (2D) ones in the dimension splitting direction. The reproducing kernel particle method (RKPM) is used to solve each 2D problem, the essential boundary conditions are imposed by penalty method, and the discretized equation is obtained from Galerkin weak form of potential problems. Finite difference method is used in the dimension splitting direction. Then, by combining a series of the equations of the RKPM for solving 2D problems, the final equation of the DSRKPM for 3D potential problems is obtained. Five example problems on regular or irregular domains are selected to show that the DSRKPM has higher computational efficiency than the RKPM and the improved element-free Galerkin method for 3D potential problems.     

Multiple-scale reproducing kernel particle methods for large deformation problems

Reproducing Kernel Particle Method (RKPM) with a built-in feature of multiresolution analysis is reviewed and applied to large deformation problems. Since the application of multiresolution RKPM to the large deformation problems is still in its early stage of development, we introduce, in this paper, the concept of a multiple-scale measure which is a extension of the linear formulations to nonlinear problems. We also propose an appropriate measure to properly detect the high-scale response of a largely deformed material. Via this technique of multilevel decomposition of a reproducing kernel function, the high-scale component of the measure is used in deriving an adaptive algorithm by simply inserting extra particles. Numerical experiments for non-linear elastic materials are performed to demonstrate the completeness of multiple-scale Reproducing Kernel Particle Method. © 1998 John Wiley & Sons, Ltd.     

A Lagrangian reproducing kernel particle method for metal forming analysis

A Meshless approach based on a Reproducing Kernel Particle Method is developed for metal forming analysis. In this approach, the displacement shape functions are constructed using the reproducing kernel approximation that satisfies consistency conditions. The variational equation of materials with loading-path dependent behavior and contact conditions is formulated with reference to the current configuration. A Lagrangian kernel function, and its corresponding reproducing kernel shape function, are constructed using material coordinates for the Lagrangian discretization of the variational equation. The spatial derivatives of the Lagrangian reproducing kernel shape functions involved in the stress computation of path-dependent materials are performed by an inverse mapping that requires the inversion of the deformation gradient. A collocation formulation is used in the discretization of the boundary integral of the contact constraint equations formulated by a penalty method. By the use of a transformation method, the contact constraints are imposed directly on the contact nodes, and consequently the contact forces and their associated stiffness matrices are formulated at the nodal coordinate. Numerical examples are given to verify the accuracy of the proposed meshless method for metal forming analysis.     

A gradient reproducing kernel collocation method for boundary value problems

The earlier work in the development of direct strong form collocation methods, such as the reproducing kernel collocation method (RKCM), addressed the domain integration issue in the Galerkin type meshfree method, such as the reproducing kernel particle method, but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of collocation points for optimal convergence. In this work, we intend to address the computational complexity in RKCM while achieving optimal convergence by introducing a gradient reproduction kernel approximation. The proposed gradient RKCM reduces the order of differentiation to the first order for solving second‐order PDEs with strong form collocation. We also show that, different from the typical strong form collocation method where a significantly large number of collocation points than the number of source points is needed for optimal convergence, the same number of collocation points and source points can be used in gradient RKCM. We also show that the same order of convergence rates in the primary unknown and its first‐order derivative is achieved, owing to the imposition of gradient reproducing conditions. The numerical examples are given to verify the analytical prediction. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiresolution reproducing kernel particle methods

Reproducing Kernel Particle Methods (RKPM) with a built-in feature of multiresolution analysis are addressed. Some fundamental concepts such as reproducing conditions, and correction function are constructed to systematize the framework of RKPM. In particular, Fourier analysis, as a tool, is exploited to further elaborate RKPM in the frequency domain. Furthermore, we address error estimation and convergence properties. We present several applications which confirm the widespread applicability of multiresolution RKPM.     

A quasi‐linear reproducing kernel particle method

Reproducing kernel particle method (RKPM) has been applied to many large deformation problems. RKPM relies on polynomial reproducing conditions to yield desired accuracy and convergence properties but requires appropriate kernel support coverage of neighboring nodes to ensure kernel stability. This kernel stability condition is difficult to achieve for problems with large particle motion such as the fragment‐impact processes that commonly exist in extreme events. A new reproducing kernel formulation with ‘quasi‐linear’ reproducing conditions is introduced. In this approach, the first‐order polynomial reproducing conditions are approximately enforced to yield a nonsingular moment matrix. With proper error control of the first‐order completeness, nearly second‐order convergence rate in L2 norm can be achieved while maintaining kernel stability. The effectiveness of this quasi‐linear RKPM formulation is demonstrated by modeling several extremely large deformation and fragment‐impact problems. Copyright © 2016 John Wiley & Sons, Ltd.     

A numerical method for the solution of two-dimensional inverse heat conduction problems

A numerical method for the solution of inverse heat conduction problems in two-dimensional rectangular domains is established and its performance is demonstrated by computational results. The present method extends Beck's⁸ method to two spatial dimensions and also utilizes future times in order to stabilize the ill-posedness of the underlying problems. The approach relies on a line approximation of the elliptic part of the parabolic differential equation leading to a system of one-dimensional problems which can be decoupled.     

Adaptive reproducing kernel particle method using gradient indicator for elasto-plastic deformation

An adaptive meshless method based on the multi-scale Reproducing Kernel Particle Method (RKPM) for analysis of nonlinear elasto-plastic deformation is proposed in this research. In the proposed method, the equivalent strain, stress, and the second invariant of the Cauchy–Green deformation tensor are decomposed into two scale components, viz., high- and low-scale components by deriving them from the multi-scale decomposed displacement. Through combining the high-scale components of strain and the stress update algorithm, the equivalent stress is decomposed into two scale components. An adaptive algorithm is proposed to locate the high gradient region and enrich the nodes in the region to improve the computational accuracy of RKPM. Using the algorithm, the high-scale components of strain and stress and the second invariant of the Cauchy–Green deformation tensor are normalized and used as criteria to implement the adaptive analysis. To verify the validity of the proposed adaptive meshless method in nonlinear elasto-plastic deformation, four case studies are calculated by the multi-scale RKPM. The patch test results show that the used multi-scale RKPM is reliable in analysis of the regular and irregular nodal distribution. The results of other three cases show that the proposed adaptive algorithm can not only locate the high gradient region well, but also improve the computational accuracy in analysis of the nonlinear elasto-plastic deformation.     

Enriched reproducing kernel particle method for piezoelectric structures with arbitrary interfaces

A formulation based on enriched reproducing kernel particle method is proposed for analysing non‐homogenous piezoelectric structures with arbitrarily shaped material interfaces. The interfaces are represented using a set of cubic spline segments, each of which is defined by the co‐ordinates and the slopes of its two end‐nodes. Special enrichment functions are introduced within the reproducing kernel particle method, in order to capture discontinuities of strain and electric fields at the material interfaces. Several validation examples are provided to demonstrate the veracity and the utility of the proposed numerical method. Copyright © 2006 John Wiley & Sons, Ltd     

A reproducing kernel particle method for meshless analysis of microelectromechanical systems

Many existing computer-aided design systems for microelectromechanical systems require the generation of a three-dimensional mesh for computational analysis of the microdevice. Mesh generation requirements for microdevices are very complicated because of the presence of mixed-energy domains. Point methods or meshless methods do not require the generation of a mesh, and computational analysis can be performed by sprinkling points covering the domain of the microdevice. A corrected smooth particle hydrodynamics approach also referred to as the reproducing kernel particle method is developed here for microelectromechanical applications. A correction function that establishes the consistency and the stability of the meshless method is derived. A simple approach combining the constraint elimination and the Lagrange multiplier technique is developed for imposition of boundary conditions. Numerical results are shown for static and dynamic analysis of microswitches and electromechanical pressure sensors. The accuracy of the meshless method is established by comparing the numerical results obtained with meshless methods with previously reported experimental and numerical data.     

On coupling of reproducing kernel particle method and boundary element method

In this paper a new coupling method to merge the reproducing kernel particle method (RKPM) and the boundary element method (BEM) is proposed. In this formulation, the whole problem domain will first be divided into two subdomains in which the RKPM and the BEM are applied respectively. A simple and direct coupling procedure is then applied to preserve the compatibility of the solution along the interface of the two subdomains. Unlike other coupling procedures suggested previously, the present coupling procedure neither requires modification of the RKPM and the BEM formulations nor the use of finite elements along the interface boundary between the two subdomains. The validity and efficiency of the coupling procedure are demonstrated by using it to solve five benchmark elastostatic stress analysis problems. In addition, a simple analysis of the a priori convergence rate of the coupling procedure is given. Preliminary assessment of the convergence characteristics of the coupling procedure is done by carrying out uniform refinements on the benchmark problems.     

A corrective smoothed particle method for boundary value problems in heat conduction

Combining the kernel estimate with the Taylor series expansion is proposed to develop a Corrective Smoothed Particle Method (CSPM). This algorithm resolves the general problem of particle deficiency at boundaries, which is a shortcoming in Standard Smoothed Particle Hydrodynamics (SSPH). In addition, the method’s ability to model derivatives of any order could make it applicable for any time‐dependent boundary value problems. An example of the applications studied in this paper is unsteady heat conduction, which is governed by second‐order derivatives. Numerical results demonstrate that besides the capability of directly imposing boundary conditions, the present method enhances the solution accuracy not only near or on the boundary but also inside the domain. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. government work and is in the public domain in the United States.     

Transient two-dimensional heat conduction problems solved by the finite element method

A finite element weighted residual process has been used to solve transient linear and non-linear two-dimensional heat conduction problems. Rectangular prisms in a space-time domain were used as the finite elements. The weighting function was equal to the shape function defining the dependent variable approximation. The results are compared in tables with analytical, as well as other numerical data. The finite element method compared favourably with these results. It was found to be stable, convergent to the exact solution, easily programmed, and computationally fast. Finally, the method does not require constant parameters over the entire solution domain.     

Admissible approximations for essential boundary conditions in the reproducing kernel particle method

In the reproducing kernel particle method (RKPM), and meshless methods in general, enforcement of essential boundary conditions is awkward as the approximations do not satisfy the Kronecker delta condition and are not admissible in the Galerkin formulation as they fail to vanish at essential boundaries. Typically, Lagrange multipliers, modified variational principles, or a coupling procedure with finite elements have been used to circumvent these shortcomings. Two methods of generating admissible meshless approximations, are presented; one in which the RKPM correction function equals zero at the boundary, and another in which the domain of the window function is selected such that the approximate vanishes at the boundary. An extension of the RKPM dilation parameter is also introduced, providing the capability to generate approximations with arbitrarily shaped supports. This feature is particularly useful for generating approximations near boundaries that conform to the geometry of the boundary. Additional issues such as degeneration of shape functions from 2D to 1D and moment matrix conditioning are also addressed. The support of this research by the Office of Naval Research (ONR) to Northwestern University is gratefully acknowledged.     

Reference temperature in two-dimensional problems of heat conduction

The problem of determining the average temperature in calculating the heat transfer between coaxial cylinders is considered. It is shown that the true value of the average temperature depends on the geometry of the heat transfer surfaces and is less than the arithmetic mean value usually employed.     

Quasi-convex reproducing kernel meshfree method

A quasi-convex reproducing kernel approximation is presented for Galerkin meshfree analysis. In the proposed meshfree scheme, the monomial reproducing conditions are relaxed to maximizing the positivity of the meshfree shape functions and the resulting shape functions are referred as the quasi-convex reproducing kernel shape functions. These quasi-convex meshfree shape functions are still established within the framework of the classical reproducing or consistency conditions, namely the shape functions have similar form as that of the conventional reproducing kernel shape functions. Thus this approach can be conveniently implemented in the standard reproducing kernel meshfree formulation without an overmuch increase of computational effort. Meanwhile, the present formulation enables a straightforward construction of arbitrary higher order shape functions. It is shown that the proposed method yields nearly positive shape functions in the interior problem domain, while in the boundary region the negative effect of the shape functions are also reduced compared with the original meshfree shape functions. Subsequently a Galerkin meshfree analysis is carried out by employing the proposed quasi-convex reproducing kernel shape functions. Numerical results reveal that the proposed method has more favorable accuracy than the conventional reproducing kernel meshfree method, especially for structural vibration analysis.