Reproducing Kernel Element Method Research Articles

Symmetric global partition polynomials for reproducing kernel elements

The reproducing kernel element method is a numerical technique that combines finite element and meshless methods to construct shape functions of arbitrary order and continuity, yet retains the Kronecker- $$\delta $$ ? property. Central to constructing these shape functions is the construction of global partition polynomials on an element. This paper shows that asymmetric interpolations may arise due to such things as changes in the local to global node numbering and that may adversely affect the interpolation capability of the method. This issue arises due to the use of incomplete polynomials that are subsequently non-affine invariant. This paper lays out the new framework for generating general, symmetric, truly minimal and complete affine invariant global partition polynomials for triangular and tetrahedral elements. Optimal convergence rates were observed in the solution of Kirchhoff plate problems with rectangular domains.

Reproducing Kernel Element Method for Galerkin Solution of Elastostatic Problems

The Reproducing Kernel Element Method (RKEM) is a relatively new technique developed to have two distinguished features: arbitrary high order smoothness and arbitrary interpolation order of the shape functions. This paper provides a tutorial on the derivation and computational implementation of RKEM for Galerkin discretizations of linear elastostatic problems for one and two dimensional space. A key characteristic of RKEM is that it do not require mid-side nodes in the elements to increase the interpolatory power of its shape functions, and contrary to meshless methods, the same mesh used to construct the shape functions is used for integration of the stiffness matrix. Furthermore, some issues about the quadrature used for integration arediscussed in this paper. Its hopes that this may attracts the attention of mathematicians.

Gradient-enhanced damage modeling of cracked bodies by reproducing kernel element method

In crack modeling by extended finite element method, the enrichment by the asymptotic analytical solution works excellent for linear elastic fracture mechanics. However for the more sophisticated condition of a gradient-enhanced damage model, firstly the analytical solution is not available and secondly higher order continuity of the shape functions is required. In this study, the extended finite element model of the cracked body is augmented with shape functions of higher order continuity and higher order consistency generated by reproducing kernel element method. They are used in a region near the crack tip to have an appropriate interpolation of the damaged zone while for the rest of the solution domain standard shape functions are used to reduce the computational cost. No new nodes are needed when defining the higher order shape functions by reproducing kernel element method and thus remeshing is avoided during the crack growth. After describing the formulations, several numerical examples are implemented to investigate the ability of the proposed method, for both the linear elastic fracture mechanics and the gradient-enhanced damage models.

The quasi-uniformity condition for reproducing kernel element method meshes

The reproducing kernel element method is a hybrid between finite elements and meshfree methods that provides shape functions of arbitrary order and continuity yet retains the Kronecker-δ property. To achieve these properties, the underlying mesh must meet certain regularity constraints. This paper develops a precise definition of these constraints, and a general algorithm for assessing a mesh is developed. The algorithm is demonstrated on several mesh types. Finally, a guide to generation of quasi-uniform meshes is discussed.

Geometry representation, modification and iterative design using RKEM

We present a new method for representing smooth geometry using the reproducing kernel element method. Further, the formulation provides a simple method to modify geometry that does not require modification of the underlying RKEM mesh. We develop a method for determining the representation from a discrete point set that is applicable to both traditional engineering domains, as well as applications where only discrete point sets are available, such as biology and medicine. Geometry representation and modification are demonstrated using examples of analytic functions and on a digitized shark tooth. Finally, we develop an isogeometric analysis capability for modified and un-modified geometry. The isogeometric capability on modified geometry may be useful in an iterative design-analyze cycle or shape optimization.

Treatment of discontinuity in the reproducing kernel element method

AbstractA discontinuous reproducing kernel element approximation is proposed in the case where weak discontinuity exists over an interface in the physical domain. The proposed method can effectively take care of the discontinuity of the derivative by truncating the window function and global partition polynomials. This new approximation keeps the advantage of both finite element methods and meshfree methods as in the reproducing kernel element method. The approximation has the interpolation property if the support of the window function is contained in the union of the elements associated with the corresponding node; therefore, the continuity of the primitive variables at nodes on the interface is ensured. Furthermore, it is smooth on each subregion (or each material) separated by the interface. The major advantage of the method is its simplicity in implementation and it is computationally efficient compared to other methods treating discontinuity. The convergence of the numerical solution is validated through calculations of some material discontinuity problems. Copyright © 2005 John Wiley & Sons, Ltd.

Microforming: Experimental Investigation of the Extrusion Process for Micropins and its Numerical Simulation Using RKEM

Microforming using a small machine (or so-called desktop machine) is an alternative new approach to those using full-size heavy equipment for manufacturing microparts. Microparts are commonly defined as parts or structures with at least two dimensions in the submillimeter range, which are used extensively in electronics and micromechanical products. However, when scaling down a conventional forming process to microscale, the influence of the so-called size effect needs to be considered. The individual microstructure (size, shape, and orientation of grains) and the interfacial conditions show a significant effect on the process characteristics. In this paper, the process of extrusion is investigated to establish it as a viable process for microforming. A forming assembly is fabricated and used in conjunction with a loading substage to extrude micropins with a final diameter of 1 mm. The effect of grain size is investigated by using workpieces heat treated to produce grain sizes varying from 32 μm up to 211 μm. Two extrusion dies with different roughness are used to study the effect of surface finish. While experiments lead to interesting questions and new discoveries, theoretical or numerical solutions are necessary tools for process optimization. Here, knowing the limits of the current widely used numerical simulation tools [i.e., the Finite Element Method (FEM)], a new method, the Reproducing Kernel Element Method (RKEM), has recently been developed to address the limitations of the FEM (for example, remeshing issue), while maintaining FEM’s advantages, e.g., the polynomial reproducing property and function interpolation property. The new RKEM method is used to simulate the microextrusion problem. Its results are compared with that obtained from the FEM and the experiment result. Satisfactory results were obtained. Future directions on the experimental and simulation work are addressed.

Reproducing kernel element method Part II: Globally conforming Im/ Cn hierarchies

In this part of the work, a minimal degrees of freedom, arbitrary smooth, globally compatible, I m / C n interpolation hierarchy is constructed in the framework of reproducing kernel element method (RKEM) for arbitrary multiple dimensional domains. This is the first interpolation hierarchical structure that has been constructed with both minimal degrees of freedom and higher order smoothness or continuity over multi-dimensional domain. The proposed hierarchical structure possesses the generalized Kronecker property, i.e. ∂ αΨ (β) I/ ∂x α(x J)=δ IJδ αβ, |α|,|β|⩽m. This contribution is the latest breakthrough of an outstanding problem––construction of a minimal degrees of freedom, globally conforming, I m / C n finite element interpolation fields on an arbitrary mesh or subdivision of multiple dimension. The newly constructed globally conforming interpolant is a hybrid of a set of C ∞ global partition polynomials with a highly smooth ( C n ) compactly supported meshfree partition of unity. Examples of compatible RKEM hierarchical interpolations are illustrated, and they are used in a Galerkin procedure to solve differential equations.

Reproducing kernel element method Part III: Generalized enrichment and applications

In this part of the work, a notion of generalized enrichment is proposed to construct the global partition polynomials or to enrich global partition polynomial basis with extra terms corresponding to the higher order derivatives of primary variable. This is accomplished by either multiplying enrichment functions with the original global partition polynomials, or increasing the order of global partition polynomials in the same mesh. Without refining mesh, high order consistency in interpolation hierarchy with generalized Kronecker delta property can be straightforwardly achieved in quadrilateral and triangular mesh in 2D by the proposed scheme. Comparing with the traditional finite element methods, the construction proposed here has more flexibility and only needs minimal degrees of freedom. The optimal element with high reproducing capacity and overall minimal degrees of freedom can be constructed by the generalized enrichment procedure. Two optimal elements in two dimensional space have been constructed: T10P3I 4 3 triangular element satisfies third order consistency condition with only 10 degrees of freedom, and Q15P4I 4 3 quadrilateral element satisfies fourth order consistency condition with 15 degrees of freedom. The performance of interpolation hierarchy is evaluated through solving some bench-mark problems for thin (Kirchhoff) plates.

Reproducing kernel element method. Part I: Theoretical formulation

In this paper and its sequels, we introduce and analyze a new class of methods, collectively called the reproducing kernel element method (RKEM). The central idea in the development of the new method is to combine the strengths of both finite element methods (FEM) and meshfree methods. Two distinguished features of RKEM are: the arbitrarily high order smoothness and the interpolation property of the shape functions. These properties are desirable especially in solving Galerkin weak forms of higher order partial differential equations and in treating Dirichlet boundary conditions. So unlike the FEM, there is no need for special treatment with the RKEM in solving high order equations. Compared to meshfree methods, Dirichlet boundary conditions do not present any difficulty in using the RKEM. A rigorous error analysis and convergence study of the method are presented. The performance of the method is illustrated and assessed through some numerical examples.

Reproducing kernel element method. Part IV: Globally compatible [formula omitted] triangular hierarchy

In this part of the work, a globally compatible C n(Ω) triangular element hierarchy is constructed in the framework of reproducing kernel element method (RKEM) for arbitrary two dimensional domains. In principle, the smoothness of the globally conforming element can be made arbitrarily high ( n⩾1). The triangle interpolation field can interpolate the derivatives of an unknown function up to arbitrary mth order, ( I m ), and it can reproduce complete kth order polynomials with k⩾ m. This is the first interpolation hierarchical structure that has ever been constructed with both minimal degrees of freedom and higher order smoothness and continuity over discretizations of a multiple dimensional domain. The performance of the newly constructed compatible element is evaluated in solving several Kirchhoff plate problems.