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The reproducing kernel particle methods (RKPM) are meshfree methods arising in mechanics, especially in dealing with problems involving large deformation and singularities. We provide a theoretical analysis of super-convergence in Sobolev norms for reproducing kernel (RK) approximations when the interpolation order p is even. Super-convergence phenomenon means the convergence rate is higher than the order that is generally expected. We distinguish the continuous RK approximation and the discrete RKP approximation. While the continuous RK approximations are proven to be super-convergent when p is even, its discrete counterpart has super-convergence only with uniform particle distribution and special choices of RK kernel functions and support sizes. Moreover, super-convergence does not exist for the discrete RKP approximation with general RK support sizes. The concept of pseudo-super-convergence is then introduced to explain why in practice the super-convergence phenomenon is sometimes observed for general cases although in theory it is not true. Our analysis is general for multi-dimensional RK approximations.

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... In the rest of the work, we assume that the reproducing condition (3.9) is satisfied with p = 1, with which we call our method the linear RK approximation and the RK basis function is referred to as the linear RK basis. Letting \bfita = 2\bfith , then it can be shown (see, e.g., [24]) that the correction function C(\bfitx ; \bfitx -\bfitx \bfitk ) \equiv 1 and the linear RK basis function is reduced to (3.16) \Psi ...

... In general, we can choose the RK support as \bfita = 2r 0 \bfith for r 0 \in \BbbN . In this case it is shown in [24,Lemma 4.4] that \bfitphi \bfita satisfies the Strang--Fix condition [39], and therefore it can be shown that the moments are constants and they satisfy the same properties in Lemma 3.1 [24]. In the case p = 1, it also implies that the correction function C(\bfitx ; \bfitx -\bfitx \bfitk ) \equiv C for some constant C. ...

... In general, we can choose the RK support as \bfita = 2r 0 \bfith for r 0 \in \BbbN . In this case it is shown in [24,Lemma 4.4] that \bfitphi \bfita satisfies the Strang--Fix condition [39], and therefore it can be shown that the moments are constants and they satisfy the same properties in Lemma 3.1 [24]. In the case p = 1, it also implies that the correction function C(\bfitx ; \bfitx -\bfitx \bfitk ) \equiv C for some constant C. ...

... Remark 3.1. For the simplicity of presentation, we choose the RK support size a = a 0 h where a 0 = 2 in this work but the analysis works for a general even number a 0 [29,43]. ...

... Thus, the RK basis function can reproduce linear polynomials [36,43,44], i.e., ...

... Combining Eqs. (43) and (44), we have ...

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.

... Remark 3.1. For the simplicity of presentation, we choose the RK support size a = a 0 h where a 0 = 2 in this work but the analysis works for a general even number a 0 [29,43]. ...

... Thus, the RK basis function can reproduce linear polynomials [36,43,44], i.e., ...

... Combining Eqs. (43) and (44), we have ...

In this work, we study the reproducing kernel (RK) collocation method for the peridynamic Navier equation. We first apply a linear RK approximation on both displacements and dilatation, then back-substitute dilatation, and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit 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. We then 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 Lam\'{e} parameters satisfy $\lambda \geq \mu$ to avoid adding extra constraints on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.

... Remark 3.1. Choose the RK support as a = a 0 h, where a 0 is an even number, the linear RK basis function becomes a rescaling of the cubic B-spline function ( [21]) and the RK approximation has synchronized convergence for L 2 , H 1 and H 2 error norm ( [22]). For the simplicity of presentation, we assume a 0 = 2 in the paper but the analysis also works for general even number a 0 . ...

... Let a 0 = 2 be the parameter described in Remark 3.1, then the linear RK basis function can be written as ( [21]) ...

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).

... The RKPM approach is proposed based on the Galerkin method and integral transformation in order to improve SPH performance. The RKPM approach has been able to perform numerical simulation analysis for a large number of engineering problems, such as structural mechanics, fluid dynamics, and large deformation [24][25][26][27][28]. Subsequent studies show that MLS and RKPM are consistent, but different from the FEM, the approximate function constructed by these methods lacks the Kronecker-Delta property. ...

We propose a reproducing kernel particle method-based smoothed generalized finite element method (RKPM-SGFEM) for 2D and 3D structural analysis. As with partition of unity idea, the displacement function in RKPM-SGFEM is discretized as finite element shape function and local approximation, where the local approximation is obtained by Taylor truncation in nodal support domain. The gradient smoothing reproducing kernel (RK) meshfree approximation is utilized for the derivatives of Taylor polynomials. It is well known that RKPM does not possess Kronecker-Delta property. However, this defect of RKPM is suppressed under the proposed RKPM-SGFEM framework, because of the novel combination of element shape function and Taylor expansion. In addition, the final composite shape function also ensures partitions of unity, which is not affected by the meshfree shape function. Subsequently, we performed a series of numerical tests on the proposed RKPM-SGFEM, which not only passes linear independent and zero energy modal tests, but also shows higher accuracy, error convergence speed, and efficiency in numerical analysis of 2D and 3D problems. Besides, numerical verification also indicates that RKPM-SGFEM is insensitive to mesh distortion, temporally stable, and highly consistent with experimental test in practical engineering analysis.

... While several stable results have been presented for non-symmetric methods in elastodynamics [19][20][21][22][23], that authors believe these are special cases of uniform discretizations where quasi-symmetric matrices have been observed in our studies (later several demonstration problems will be presented). It is well-known by some researchers for instance, meshfree and isogeometric approximations (and discretizations in general) often result in special cases for uniform discretizations [24][25][26][27][28][29][30][31]. ...

Non-symmetric matrices may arise in the discretization of self-adjoint problems when a Petrov–Galerkin, collocation, or finite-volume method is employed. While these methods have been widely applied, in this paper it is shown that the use of these non-symmetric matrices is incompatable with the conservation of energy in elastodynamics. First, the consistency between the continuous forms of the momentum equation and the energy equation is examined. It is shown that the conservation of linear momentum is equivalent to conservation of energy provided the solution is sufficiently smooth. The semi-discrete counterparts are then analyzed, where it is demonstrated that they are also equivalent, but only conditionally: the mass and stiffness matrices must be symmetric. As a result, employing a non-symmetric method in elastodynamics may artificially generate or dissipate energy. The fully discrete forms with Newmark time integration are then examined where it is shown that unconditionally unstable algorithms may arise. An energy-conserving time integration algorithm is then proposed which provides stability in the solutions of non-symmetric systems. The collocation and finite-volume methods are employed in numerical examples to demonstrate stability issues and the effectiveness of the proposed time integration methodology.

... While several stable results have been presented for non-symmetric methods in elastodynamics [19][20][21][22][23], that authors believe these are special cases of uniform discretizations where quasi-symmetric matrices have been observed in our studies (later several demonstration problems will be presented). It has been well-known by some researchers for instance, meshfree and isogeometric approximations (and discretizations in general) can reproduce several different scenarios, particularly in special cases of uniform discretizations [24][25][26][27][28][29][30][31]. ...

Highlights • Analysis of energy-momentum consistency on continuous level and discrete level • Temporal stability analysis in solving elastodynamics by non-symmetric type methods • Unconditionally unstable solutions for Petrov-Galerkin methods shown • A time integration is proposed to exactly conserve energy in non-symmetric systems Abstract Non-symmetric matrices may arise in the discretization of elastodynamics when a Petrov-Galerkin, collocation method, or finite-volume method is employed. While these methods have been widely applied, in this paper it is shown that the use of these non-symmetric matrices is inconsistent with the conservation of energy. First, the consistency between the continuous forms of the momentum equation and energy equation is examined. It is shown that the conservation of linear momentum is equivalent to conservation of energy provided the solution is sufficiently smooth. The semi-discrete counterparts are then examined, where it is shown they are also equivalent, but only conditionally: the mass and stiffness matrices must be symmetric. As a result, employing a non-symmetric method elastodynamics may artificially generate or dissipate energy. The fully discrete forms with Newmark time integration are then examined where it is shown that unconditionally unstable algorithms may arise. An energy-conserving time integration algorithm is then proposed which provides stability non-symmetric systems. The collocation and finite-volume methods are employed in numerical examples to demonstrate the effectiveness of proposed time integration methodology.

... Fig. 5 shows the same data for the vase-like shape. For the normal vector, the quadratic discretization shows a nearly second-order convergence rate, while the cubic and quartic discretizations converge at a rate between 3.5 and 4. For the curvatures, the quadratic and cubic discretizations converge at the rate between 1.5 and 2, while the quartic discretizations converge at the rate between 3.5 and 4. The results indicate the presence of super-convergence, which is not uncommon for meshfree schemes [56,63,112,136]. In particular, when calculating the normal vector, which involves only the first-order derivatives, the super-convergence behavior happens for the odd orders (i.e., cubic). ...

We present a comprehensive rotation-free Kirchhoff–Love (KL) shell formulation for peridynamics (PD) that is capable of modeling large elasto-plastic deformations and fracture in thin-walled structures. To remove the need for a predefined global parametric domain, Principal Component Analysis is employed in a meshfree setting to develop a local parameterization of the shell midsurface. The KL shell kinematics is utilized to develop a correspondence-based PD formulation. A bond-stabilization technique is employed to naturally achieve stability of the discrete solution. Only the mid-surface velocity degrees of freedom are used in the governing thin-shell equations. 3D rate-form material models are employed to enable simulating a wide range of material behavior. A bond-associative damage correspondence modeling approach is adopted to use classical failure criteria at the bond level, which readily enables the simulation of brittle and ductile fracture. Discretizing the model with asymptotically compatible meshfree approximation provides a scheme which converges to the classical KL shell model while providing an accurate and flexible framework for treating fracture. A wide range of numerical examples, ranging from elastostatics to problems involving plasticity, fracture, and fragmentation, are conducted to validate the accuracy, convergence, and robustness of the developed PD thin-shell formulation. It is also worth noting that the present method naturally enables the discretization of a shell theory requiring higher-order smoothness on a completely unstructured surface mesh.

... Figure 5 shows the same data for the vase-like shape. For the normal vector, the quadratic discretization shows a nearly second-order convergence rate, while the cubic and quartic discretizations converge at a rate between 3.5 and 4. For the curvatures, the quadratic and cubic discretizations converge at the rate between 1.5 and 2, while the quartic dicretizations converge at the rate between 3.5 and 4. The results indicate the presence of super-convergence, which is not uncommon for meshfree schemes [65,112,50,13]. In particular, fwhen calculating the normal vector, which involves only the first-order derivatives, the super-convergence behavior happens for the odd orders (i.e., cubic). On the other hand, the super-convergence behavior shifts to the even orders (i.e., quadratic and quartic) for computing the curvatures, in which both the first-and second-order derivatives are involved. ...

We present a comprehensive rotation-free Kirchhoff-Love (KL) shell formulation for peridynam-ics (PD) that is capable of modeling large elasto-plastic deformations and fracture in thin-walled structures. To remove the need for a predefined global parametric domain, Principal Component Analysis is employed in a meshfree setting to develop a local parameterization of the shell mid-surface. The KL shell kinematics is utilized to develop a correspondence-based PD formulation. A bond-stabilization technique is employed to naturally achieve stability of the discrete solution. Only the mid-surface velocity degrees of freedom are used in the governing thin-shell equations. 3D rate-form material models are employed to enable simulating a wide range of material behavior. A bond-associative damage correspondence modeling approach is adopted to use classical failure criteria at the bond level, which readily enables the simulation of brittle and ductile fracture. Discretizing the model with asymptotically compatible meshfree approximation provides a scheme which converges to the classical KL shell model while providing an accurate and flexible framework for treating fracture. A wide range of numerical examples, ranging from elastostatics to problems involving plasticity, fracture, and fragmentation, are conducted to validate the accuracy, convergence, and robustness of the developed PD thin-shell formulation. It is also worth noting that the present method naturally enables the discretization of a shell theory requiring higher-order smoothness on a completely unstructured surface mesh.

... The effect of varying normalized support sizes in the proposed method is now examined, since as shown previously, increased support sizes in the RK approximation can yield different behavior on the essential boundary of the domain for both test and trial functions. In addition, the present test is to show that the previous results were not a special case-window functions and their measure can have an effect on accuracy and convergence rates [25], and even superconvergence can be obtained for special values of window functions [22,23]. Thus the current permutations on a and p will examine the robustness of the formulation under the variety of free parameters in the RK approximation. ...

Enforcement of essential boundary conditions in many Galerkin meshfree methods is non-trivial due to the fact that field variables are not guaranteed to coincide with their coefficients at nodal locations. A common approach to overcome this issue is to strongly enforce the boundary conditions at these points by employing a technique to modify the approximation such that this is possible. However, with these methods, test and trial functions do not strictly satisfy the requirements of the conventional weak formulation of the problem, as the desired imposed values can actually deviate between nodes on the boundary. In this work, it is first shown that this inconsistency results in the loss of Galerkin orthogonality and the best approximation property, and correspondingly, failure to pass the patch test. It is also shown that this induces an O(h) error in the energy norm in the solution of second-order boundary value problems that is independent of the order of completeness in the approximation. As a result, this places a barrier on the global order of accuracy of Galerkin meshfree solutions to that of linear consistency. That is, with these methods, it is not possible to attain the higher order accuracy offered by meshfree approximations in the solution of boundary-value problems. To remedy this deficiency, two new weak forms are introduced that relax the requirements on the test and trial functions in the traditional weak formulation. These are employed in conjunction with strong enforcement of essential boundary conditions at nodes, and several benchmark problems are solved to demonstrate that optimal accuracy and convergence rates associated with the order of approximation can be restored using the proposed method. In other words, this approach allows p-refinement, and h-refinement with pth order rates with strong enforcement of boundary conditions beyond linear (p>1) for the first time. In addition, a new concept termed a-refinement is introduced, where improved accuracy is obtained by increasing the kernel measure in meshfree approximations, previously unavailable.

... It should be noted that the linear and quadratic results are similar in the uniform case (and to some extent in the non-uniform case). Super-convergence of the solution in some cases is observed, i.e., observing the rate of n + 1 for an nth-order method, which agrees with the results of [15,17,19,35]. In this example, not only for the linear case, but also for the quadratic case (due to super-convergence), the BA-PD and BA-RK-PD approaches produce the same results. ...

The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make use of higher-order corrections to improve the computation of integrals in the correspondence formulation. A unified approach is presented that incorporates the reproducing kernel (RK) and generalized moving least square (GMLS) approximations in PD to obtain non-local gradients of higher-order accuracy. We show, however, that the improved quadrature rule does not suffice to handle instability issues that have proven problematic for the correspondence model-based PD. In Part I of this paper, a bond-associative, higher-order core formulation is developed that naturally provides stability without introducing artificial stabilization parameters. Numerical examples are provided to study the convergence of RK-PD, GMLS-PD, and their bond-associated versions to a local counterpart, as the degree of non-locality (i.e., the horizon) approaches zero. Problems from linear elastostatics are utilized to verify the accuracy and stability of our approach. It is shown that the bond-associative approach improves the robustness of RK-PD and GMLS-PD formulations, which is essential for practical applications. The higher-order, bond-associated model can obtain second-order convergence for smooth problems and first-order convergence for problems involving field discontinuities, such as curvilinear free surfaces. In Part II of this paper we use our unified PD framework to (a) study wave propagation phenomena, which have proven problematic for the state-based correspondence PD framework; (b) propose a new methodology to enforce natural boundary conditions in correspondence PD formulations, which should be particularly appealing to coupled problems. Our results indicate that bond-associative methods accompanied by higher-order gradient corrections provide the key ingredients to obtain the necessary accuracy, stability, and robustness characteristics needed for engineering-scale simulations.

... The effect of varying normalized support sizes in the proposed method is now examined, since as shown previously, increased support sizes in the RK approximation can yield different behavior on the essential boundary of the domain for both test and trial functions. In addition, the present test is to show that the previous results were not a special case-window functions and their measure can have an effect on accuracy and convergence rates [25], and even superconvergence can be obtained for special values of window functions [22,23]. Thus the current permutations on a and p will examine the robustness of the formulation under the variety of free parameters in the RK approximation. ...

Highlights
• Two weak forms are introduced that are consistent with meshfree approximations
• Higher order optimal h-refinement previously unavailable
• p-refinement previously unavailable
• New ability to increase accuracy called a-refinement
Abstract
Enforcement of essential boundary conditions in many Galerkin meshfree methods is non-trivial due to the fact that field variables are not guaranteed to coincide with their coefficients at nodal locations. A common approach to overcome this issue is to strongly enforce the boundary conditions at these points by employing a technique to modify the approximation such that this is possible. However, with these methods, test and trial functions do not strictly satisfy the requirements of the conventional weak formulation of the problem, as the desired imposed values can actually deviate between nodes on the boundary. In this work, it is first shown that this inconsistency results in the loss of Galerkin orthogonality and best approximation property, and correspondingly, failure to pass the patch test. It is also shown that this induces an O(h) error in the energy norm in the solution of second-order boundary value problems that is independent of the order of completeness in the approximation. As a result, this places a barrier on the global order of accuracy of Galerkin meshfree solutions to that of linear consistency. That is, with these methods, it is not possible to attain the higher order accuracy offered by meshfree approximations in the solution of boundary-value problems. To remedy this deficiency, two new weak forms are introduced that relax the requirements on the test and trial functions in the traditional weak formulation. These are employed in conjunction with strong enforcement of essential boundary conditions at nodes, and several benchmark problems are solved to demonstrate that optimal accuracy and convergence rates associated with the order of approximation can be restored using the proposed method. In other words, this approach allows p-refinement, and h-refinement with p th order rates with strong enforcement of boundary conditions beyond linear (p > 1) for the first time. In addition, a new concept termed a-refinement is introduced, where improved accuracy is obtained by increasing the kernel measure in meshfree approximations, previously unavailable.

... It should be noted that the linear and quadratic results are similar in the uniform case (and to some extent in the non-uniform case). Super-convergence of the solution in some cases is observed, i.e., observing the rate of n + 1 for an n th -order method, which agrees with the results of [17,35,15,19]. In this example, not only for the linear case, but also for the quadratic case (due to super-convergence), the BA-PD and BA-RK-PD approaches produce the same results. ...

The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make use of higher-order corrections to improve the computation of integrals in the correspondence formulation. A unified approach is presented that incorporates the reproducing kernel (RK) and generalized moving least square (GMLS) approximations in PD to obtain higher-order non-local gradients. We show, however, that the improved quadrature rule does not suffice to handle instability issues that have proven problematic for the correspondence-model based PD. In Part I of this paper, a bond-associative, higher-order core formulation is developed that naturally provides stability without introducing artificial stabilization parameters. Numerical examples are provided to study the convergence of RK-PD, GMLS-PD, and their bond-associated versions to a local counterpart, as the degree of non-locality (i.e., the horizon) approaches zero. Problems from linear elastostatics are utilized to verify the accuracy and stability of our approach. It is shown that the bond-associative approach improves the robustness of RK-PD and GMLS-PD formulations, which is essential for practical applications. The higher-order, bond-associated model can obtain second-order convergence for smooth problems and first-order convergence for problems involving field discontinuities, such as curvilinear free surfaces. In Part II of this paper we use our unified PD framework to: (a) study wave propagation phenomena, which have proven problematic for the state-based correspondence PD framework; (b) propose a new methodology to enforce natural boundary conditions in correspondence PD formulations, which should be particularly appealing to coupled problems. Our results indicate that bond-associative formulations accompanied by higher-order gradient correction provide the key ingredients to obtain the necessary accuracy, stability, and robustness characteristics needed for engineering-scale simulations.

... The displacement RMS errors are shown in Fig. 5. For the uniform grids, the RK-PD and GMLS-PD results show a super-convergence behavior, where a rate of convergence of n + 1 is empirically observed for odd-order models [17]. For the noted models, an inconsistent behavior is observed for the non-uniform case, which is rooted in the lack of stability. ...

The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make use of higher-order corrections to improve the computation of integrals in the correspondence formulation. A unified approach is presented that incorporates the reproducing kernel (RK) and generalized moving least square (GMLS) approximations in PD to obtain higher-order non-local gradients. We show, however, that the improved quadrature rule does not suffice to handle instability issues that have proven problematic for the correspondence-model based PD. In Part I of this paper, a bond-associative, higher-order core formulation is developed that naturally provides stability without introducing artificial stabilization parameters. Numerical examples are provided to study the convergence of RK-PD, GMLS-PD, and their bond-associated versions to a local counterpart, as the degree of non-locality (i.e., the horizon) approaches zero. Problems from linear elastostatics are utilized to verify the accuracy and stability of our approach. It is shown that the bond-associative approach improves the robustness of RK-PD and GMLS-PD formulations, which is essential for practical applications. The higher-order, bond-associated model can obtain second-order convergence for smooth problems and first-order convergence for problems involving field discontinuities, such as curvilinear free surfaces. In Part II of this paper we use our unified PD framework to: (a) study wave propagation phenomena, which have proven problematic for the state-based correspondence PD framework; (b) propose a new methodology to enforce natural boundary conditions in correspondence PD formulations, which should be particularly appealing to coupled problems. Our results indicate that bond-associative formulations accompanied by higher-order gradient correction provide the key ingredients to obtain the necessary accuracy, stability, and robustness characteristics needed for engineering-scale simulations.

In a Hilbert space H, discrete families of vectors {hj} with the property that f=∑j〈hj‖ f〉hj for every f in H are considered. This expansion formula is obviously true if the family is an orthonormal basis of H, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non-Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.

A nonlinear formulation of the Reproducing Kernel Particle Method (RKPM) is presented for the large deformation analysis
of rubber materials which are considered to be hyperelastic and nearly incompressible. In this approach, the global nodal
shape functions derived on␣the basis of RKPM are employed in the Galerkin approximation of the variational equation to formulate
the discrete equations of a boundary-value hyperelasticity problem. Existence of a solution in RKPM discretized hyperelasticity
problem is discussed. A Lagrange multiplier method and a direct transformation method are presented to impose essential boundary
conditions. The characteristics of material and spatial kernel functions are discussed. In the present work, the use of a
material kernel function assures reproducing kernel stability under large deformation. Several of numerical examples are presented
to study the characteristics of RKPM shape functions and to demonstrate the effectiveness of this method in large deformation
analysis. Since the current approach employs global shape functions, the method demonstrates a superior performance to the conventional finite element methods in dealing
with large material distortions.

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.

In this paper, we study a flexible piecewise approximation technique based on the use of the idea of the partition of unity. The approximations are piecewisely defined, globally smooth up to any order, enjoy polynomial reproduc- ing conditions, and satisfy nodal interpolation conditions for function values and derivatives of any order. We present various properties of the approximations, that are desirable properties for optimal order convergence in solving boundary value problems.

The most common discretization method for peridynamic models used in engineering problems is the node-based meshfree approach. This method discretizes peridynamic domains by a set of nodes, each associated with a nodal cell with a characteristic volume, leading to a particle-based description of continuum systems. The behavior of each particle is then considered representative of its cell. This limits the convergence rate to the first order. In this paper, we introduce a reproducing kernel (RK) approximation to the field variables in the peridynamic equations to increase the order of convergence of peridynamic numerical solutions. The numerical results demonstrate improved convergence rates in static peridynamic problems using the proposed method.

A new version of the Strang-Fix conditions is formulated and it is used to give a new proof for the characterization of the local approximation order of the spaces generated by a finite number of compactly supported basis functions and their shifts.

This paper formulates the moving least-square interpolation scheme in a framework of the so-called moving least-square reproducing kernel (MLSRK) representation. In this study, the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation, which, as a generalization of the early discrete approach, establishes a continuous basis for a partition of unity. This new formulation possesses the quality of simplicity, and it is easy to implement. Moreover, the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution, but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy.In this contribution, a generic m-consistency relation has been found, which is the essential property of the MLSRK approximation. An interpolation error estimate is given to assess the convergence rate of the approximation. It is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the Sobolev norms. As a meshless method, the convergence rate is measured by a new control variable—dilation parameter ρ of the window function, instead of the mesh size h as usually done in the finite element analysis. To illustrate the procedure, convergence has been shown for the numerical solution of the second-order elliptic differential equations in a Galerkin procedure invoked with this interpolant. In the numerical example, a two point boundary problem is solved by using the method, and an optimal convergence rate is observed with respect to various norms.

In Part I of this work, the moving least-square reproducing kernel (MLSRK) method is formulated and implemented. Based on its generic construction, an m-consistency structure is discovered and the convergence theorems are established. In this part of the work, a systematic Fourier analysis is employed to evaluate and further establish the method. The preliminary Fourier analysis reveals that the MLSRK method is stable for sufficiently dense, non-degenerated particle distribution, in the sense that the kernel function family satisfies the Riesz bound. One of the novelties of the current approach is to treat the MLSRK method as a variant of the ‘standard’ finite element method and depart from there to make a connection with the multiresolution approximation. In the spirits of multiresolution analysis, we propose the following MLSRK transformation, The highlight of this paper is to embrace the MLSRK formulation with the notion of the controlled fp-approximation. Based on its characterization, the Strang-Fix condition for example, a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the MLSRK algorithm. The main effort here is to obtain a constant correction function in the interior region of a general domain, i.e. ρh = 1. This can create a leap in the approximation order of the MLSRK algorithm significantly, if a highly smooth window function is embedded within the kernel. One consequence of this development is the synchronized convergence phenomenon—a unique convergence mechanism for the MLSRK method, i.e. by properly tuning the dilation parameter, the convergence rate of higher-order error norms will approach the same order convergence rate of the L2 error norm—they are synchronized.

SUMMARY This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the dierential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with dierent combinations of the hierarchical kernels, a synchronized convergence eect may be observed. Being dierent from the conventional Legendre function based p-type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels form a 'partition of nullity'. These newly developed kernels can be used as the multi- scale basis to solve partial dierential equations in numerical computation as a p-type renement. Copyright ? 1999 John Wiley & Sons, Ltd.

The explicit Reproducing Kernel Particle Method (RKPM) is presented and applied to the simulations of large deformation problems. RKPM is a meshless method which does not need a mesh structure in its formulation. Because of this mesh-free property, RKPM is able to simulate large deformation problems without remeshing which is often required for the mesh-based methods such as the finite element method. The RKPM shape function and its derivatives are constructed by imposing the consistency conditions. An efficient treatment of essential boundary conditions is also proposed for explicit time integration. The Lagrangian method based on the reference configuration is employed for the RKPM simulation of large deformation problems. Several examples of non-linear elastic materials are solved to demonstrate the performance of the method. The numerical experiment for the problem of underwater bubble explosion is also performed using the explicit Lagrangian RKPM formulation. © 1998 John Wiley & Sons, Ltd.

A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time-frequency and space-wave number localization of a window function is developed. This method is motivated by the theory of wavelets and also has the desirable attributes of the recently proposed smooth particle hydrodynamics (SPH) methods, moving least squares methods (MLSM), diffuse element methods (DEM) and element-free Galerkin methods (EFGM). The proposed method maintains the advantages of the free Lagrange or SPH methods; however, because of the addition of a correction function, it gives much more accurate results. Therefore it is called the reproducing kernel particle method (RKPM). In computer implementation RKPM is shown to be more efficient than DEM and EFGM. Moreover, if the window function is C∞, the solution and its derivatives are also C∞ in the entire domain. Theoretical analysis and numerical experiments on the 1D diffusion equation reveal the stability conditions and the effect of the dilation parameter on the unusually high convergence rates of the proposed method. Two-dimensional examples of advection-diffusion equations and compressible Euler equations are also presented together with 2D multiple-scale decompositions.

In this paper, a new partition of unity – the synchronized reproducing kernel (SRK) interpolant – is derived. It is a class of meshless shape functions that exhibit synchronized convergence phenomenon: the convergence rate of the interpolation error of the higher order derivatives of the shape function can be tuned to be that of the shape function itself. This newly designed synchronized reproducing kernel interpolant is constructed as an series expansion of a scaling function kernel and the associated wavelet functions. These wavelet functions are constructed in a reproducing procedure, simultaneously with the scaling function kernel, by directly enforcing certain orders of vanishing moment conditions. To the authors knowledge, this unique interpolant is the first of its kind to be constructed, and to be used in numerical computations, both in concept and in practice. The new interpolants are in fact a group of special hierarchial meshless bases, and similar counterparts may exist in spline interpolation method, other meshless methods, Galerkin-wavelet method, as well as the finite element method.
A detailed account of the subject is presented, and the mathematical principle behind the construction procedure is further elaborated. Another important discovery of this study is that the 1st order wavelet together with the scaling function kernel can be used as a weighting function in Petrov-Galerkin procedures to provide a stable numerical computation in some pathological problems. Benchmark problems in advection-diffusion problems, and Stokes flow problem are solved by using the synchronized reproducing kernel interpolant as the weighting function. Reasonably good results have been obtained. This may open the door for designing well behaved Galerkin procedures for numerical computations in various constrained media.

Large deformation analysis of non-linear elastic and inelastic structures based on Reproducing Kernel Particle Methods (RKPM) is presented. The method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties in large deformation analysis. The current formulation considers hyperelastic and elasto-plastic materials since they represent path-independent and path-dependent material behaviors, respectively. In this paper, a material kernel function and an RKPM material shape function are introduced for large deformation analysis. The support of the RKPM material shape function covers the same set of particles during material deformation and hence no tension instability is encountered in the large deformation computation. The essential boundary conditions are introduced by the use of a transformation method. The transformation matrix is formed only once at the initial stage if the RKPM material shape functions are employed. The appropriate integration procedures for the moment matrix and its derivative are studied from the standpoint of reproducing conditions. In transient problems with an explicit time integration method, the lumped mass matrices are constructed at nodal coordinate so that masses are lumped at the particles. Several hyperelasticity and elasto-plasticity problems are studied to demonstrate the effectiveness of the method. The numerical results indicated that RKPM handles large material distortion more effectively than finite elements due to its smoother shape functions and, consequently, provides a higher solution accuracy under large deformation. Unlike the conventional finite element approach, the nodal spacing irregularity in RKPM does not lead to irregular mesh shape that significantly deteriorates solution accuracy. No volumetric locking is observed when applying non-linear RKPM to nearly incompressible hyperelasticity and perfect plasticity problems. Further, model adaptivity in RKPM can be accomplished simply by adding more points in the highly deformed areas without remeshing.

Interest in meshfree (or meshless) methods has grown rapidly in recent years in solving boundary value problems (BVPs) arising in mechanics, especially in dealing with difficult problems involving large deformation, moving discontinuities, etc. In this paper, we provide a theoretical analysis of the reproducing kernel particle method (RKPM), which belongs to the family of meshfree methods. One goal of the paper is to set up a framework for error estimates of RKPM. We introduce the concept of a regular family of particle distributions and derive optimal order error estimates for RKP interpolants on a regular family of particle distributions. The interpolation error estimates can be used to yield error estimates for RKP solutions of BVPs.

This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the differential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with different combinations of the hierarchical kernels, a synchronized convergence effect may be observed. Being different from the conventional Legendre function based p- type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels fo...

Polynomial approximation of functions in Sobolev spaces

- Dupont