Numerical modeling of composite solids using an immersed meshfree Galerkin method
Introduction
The elasticity of composite solids such as particle and fiber-reinforced composites is a boundary value problem (BVP) that can be modeled by the elliptic equations containing discontinuous coefficients [1]. The discontinuous coefficients in BVPs introduce jump conditions across the interface in displacements as well as in flux. These jump conditions generally are determined by the relevant physics. The elasticity interface problems of composite solids presented in this paper consider the homogenous jump conditions which arise in a wide range of mathematical modeling in material science and bio-medical applications such as the elastic analysis of rubber compounds [2], [3], modeling of bone structures [4] and simulation of brain shifting [5]. The conventional finite element method (FEM) in this context is to generate a matching (conforming) mesh across the material interface and use standard finite element shape functions to approximate solutions of the BVPs. As a result, each element basically contains only one material and the element basis functions are independent of the jump conditions. However, generating 2D and 3D matching meshes suitable for the finite element analysis is difficult in cases where interfaces are present in irregular geometries. Construction of matching meshes in interface problem generally requires substantial user interaction and is time-consuming. Therefore from user’s point of view, it is advantageous to use discretization that is not matching at the material interfaces for the composite solids analysis.
A flexible way to couple mismatching meshes across the interface is to use the mortar finite element method [6]. It is a domain decomposition technique that enforces the jump conditions across the interface by Lagrange multipliers and results in a saddle point problem which requires appropriate solvers. Unfortunately, arbitrary choice of approximation space for Lagrange multipliers may violate the inf–sup condition [7], [8] and can lead to instabilities that eventually cause artificial oscillations in the traction fields. A different variational approach for the discretization of interface problem is offered by Nitsche’s method [9]. Such discretization incorporates interface conditions approximately, but consistently, by penalizing the jump of the primal variable (displacement) and avoids the use of Lagrange multipliers. Consequently, the resulting linear system is positive definite and does not suffer from ill-conditioning. The Nitsche’s method has a close relationship [10] with Barbosa and Hughes’ least-squares stabilized Lagrange multiplier formulation [11] in circumventing the inf–sup condition. Despite its non-trivial implementation and a need to determine the penalty parameters, Nitsche’s method has been shown to preserve optimal convergence in L2 and energy norms for elasticity interface problems [12].
Alternatively, the partition of unity method (PUM) [13], [14] employs a priori knowledge of the solution at interfaces to obtain special finite element spaces. Later in [15], the method was referred to as generalized finite element method (GFEM), since the classical FEM is a special case of this method [16]. The ability to choose a wide variety of enrichment functions in GFEM allows it to approximate non-smooth solutions of BVPs on domains having internal boundaries, corners or multiple cracks [2], [17]. The enrichment function is extrinsic to the finite element basis function, thereby introducing new degrees of freedom. In other words, the enrichment function can, in principle, be arbitrary and are certainly not limited to polynomials. This advantage, however, is achieved at a high computational cost due to expensive numerical integration [16]. The extended finite element method (XFEM) [18] is an application of PUM for problems of interface tracking and crack growth. XFEM enriches the standard finite element shape functions with additional continuous enrichment function (Ramp function) or discontinuous enrichment function (Heaviside function or Step function) to approximate the solution near the interface. The enrichment function exists only at the nodes of the elements that intersect the interface. This allows XFEM to accommodate elements that do not conform to the interface. Solution spaces of XFEM using discontinuous enrichment function do not generally satisfy the Dirichlet jump condition on the interface. As the one in the mortar finite element method, linear or tied constraint methods such as Lagrangian multipliers, penalty methods and bubble-stabilized method [19] are often utilized to enforce the Dirichlet jump condition. While such methods provide high fidelity predictions of the structure response in composites, their robustness has not yet been demonstrated on large-scale simulations of composite solids [20] such as heterogeneous microstructures. The calculations in XFEM involving small inclusions require the generation of small elements thus very time consuming. Even with high quality adaptive meshes, the nodal enrichment procedure in single element containing multiple interfaces is a significant challenge in particular for the three-dimensional problem.
Both GFEM and XFEM treat the material interface implicitly using the concept of level set method. Similar implicit boundary representation techniques have been developed based on a uniform Cartesian mesh under different versions of embedded or fictitious domain approaches. Among them are finite cell method for geometrical modeling of embedded problems [21], immersed finite element method for fluid-structure interaction problems [22] and immersed finite element method for elasticity interface problems [23], [24]. The recently developed immersed finite element method (IFEM) [23] is also rooted in the GFEM. In IFEM, the finite element basis functions are constructed to satisfy the homogenous jump conditions across the material interface whereas the mesh itself can be independent of the interface. Since the homogenous jump conditions are only satisfied point-wise at the intersections of interface and element edge, the finite element basis functions defined in this way could be discontinuous across edges of interface elements [25] and thus non-conforming. A sophisticated conforming local basis functions has been introduced to non-conforming IFEM by enlarging the support of some standard basis functions in the non-conforming finite element space such that the continuity condition in displacements can be enforced. The conforming linear IFEM [25] was shown to be optimally convergent in L2 and energy norms. Despite its improved accuracy, conforming IFEM is far more complex than non-conforming IFEM and the extension to three-dimensional case is not clear.
In comparison to finite element method, meshfree method is known to provide a higher accuracy in solid analysis in particular for the large deformation problems. The analysis of elasticity interface problem in composite solids is a new area in recent meshfree developments [26], [27], [28]. Cordes and Moran [29] impose the jump condition by adding an interface constraint in the variational formulation and solve the equation by Lagrange multiplier method. Since the material interface serves as a visibility criterion for the construction of meshfree approximation in the composite solid model, a set of interface nodes has to be manually added along the interface together with properly adjusted integration cells for the domain integration. A parallel research [30], [31] is devoted to the development of a particular meshfree approximation that contains discontinuities in the derivatives across the material interface. This approach requires defining the interface nodes in the numerical model and may not be easily performed when the interface involves complex geometry or three-dimensional object. The introduction of discontinuities to meshfree approximation has also recently been applied to the immersed particle method [32] to model the fluid–structure interaction problem. A special jump functions was also incorporated with meshfree approximation by enforcing its consistency conditions [33] and therefore additional unknowns are not needed. Recently, a discontinuous Galerkin meshfree formulation was proposed by Wang et al. [34] based on an incompatible patch-based meshfree approximation to model the material interface in composite solid. Since the continuities of displacement and normal flux are imposed weakly at the variational level, this formulation also does not acquire additional degrees of freedom.
On the other hand, conventional meshfree approximation requires special considerations to impose the boundary conditions. Wu and Koishi [3] proposed a convex generalized meshfree approximation to simplify the treatment of boundary conditions in meshfree method and applied to the micromechanical analysis of particle-reinforced rubber. The employment of convex generalized meshfree approximation to the solid analysis was shown to be less sensitive to the nodal support size [35]. The convex generalized meshfree approximation was also shown to be more robust than the non-convex meshfree approximation such as moving least-squares (MLSs) [36] or reproducing kernel (RK) [37] approximations when low-order Gaussian quadrature rule is used for numerical integration. Park et al. [38] embarked on a detailed eigenanalysis for meshfree convex approximation and reported that a large critical time step can be used in the explicit dynamic analysis. Their dispersion analysis results also reveal that meshfree convex approximation exhibits smaller lagging phase and amplitude errors than conventional MLS approximation in the full-discretization of the wave equation. Other convex meshfree approximations [39], [40] based on Shannon’s entropy concept [41] and maximum entropy [42] principle were also developed and used to solve the incompressible problem [43] as well as the structural problem [44]. The convex generalized meshfree approximation was recently introduced [45] to the standard finite element method to solve the volumetric locking problems that present in the low-order displacement-based Galerkin method. More recently, this new technique was employed to the micromechanical analysis of three-dimensional particle-reinforced rubber composite in automotive tire applications [46]. Despite its success in resolving the volumetric locking problem, the new method inevitably depends on a conforming mesh in modeling the composite solid for the micromechanical analysis. Modeling material interface in composite solid using existing meshfree techniques still relies on a mesh generation of base material and inclusions for the domain integration. The generation of non-overlapping meshes for composite solids in particular for the fiber and particular reinforced composites remains challenging in both finite element and meshfree methods.
The aim of this paper is to present an alternative approach for elasticity of composite solid problem using meshfree method. A significant feature of our approach is its flexibility to adopt the overlapping meshes in composite solid that can be easily discretized using the finite element model. Therefore the discretized domain for the base material does not need to conform to the material interface. As a consequence, there is no need to insert the interface nodes and their corresponding integration cells in the proposed method. All information regarding the geometry of material interface is stored in the geometrical model of inclusions, and solid mesh generation for inclusions can be as detail as possible to better represent the geometry of material interface. The mesh density of the inclusions can be independent of that in the base material in which a structured mesh can be easily generated for the base material. The resulting immersed meshfree method is attractive in the large-scale micromechanical modeling of composites since the periodic nodes along the boundary can be easily imposed through the structured mesh. In conforming FEM, additional efforts such as manual correction of mesh topology or employment of Fast Fourier Transformation method [20] have to be made to overcome the difficulty of constructing large meshes with homologous nodes at the periodic boundary. The proposed method is also particularly convenient in the material processing contexts when the morphology design is subjected to a change. The remainder of the paper is outlined as follows: In the next section, we define the elliptic boundary-value problem containing discontinuous coefficients and formulate the weak form of the equilibrium equation. In Section 3, we modify the variational formulation for the meshfree method to be used in the composite solid problem. The details of numerical discretization, approximation and domain integration for the modified variational method are described. A priori estimate is derived for the error measures in the energy norm. In Section 4, the numerical procedures of the immersed meshfree method are described. The discrete equations using the convex generalized meshfree approximation are presented in the same section. In Section 5, three two-dimensional numerical examples are studied. Final remarks are drawn in Section 6.
Section snippets
Problem description and variational equation
We consider an elastic solid occupying a bounded and open domain Ω ⊂ R2 with Lipschitz boundary. Let ∂ΩD and ∂ΩN be two open subsets of boundary ∂Ω such that ∂Ω = ∂ΩD ∪ ∂ΩN and ∂ΩD ∩ ∂ΩN = 0. g(x) is the prescribed displacement applied on the Dirichlet boundary ∂ΩD, and t(x) ∈ L2(∂ΩN) is the prescribed traction applied on the Neumann boundary ∂ΩN with n0 denoting the outward unit normal to the boundary ∂ΩN. The elastic body is composed of two perfectly bounded materials with zero-thickness interface Γ. The
Immersed meshfree method for composite solids
This section first focuses on the development of a new meshfree discretization that spans across the material interface. The definition of computation domain for the proposed method and the construction of the meshfree approximation that satisfies a point-wise continuity across the interface are subsequently described. Finally, a primal problem which is equivalent to a degenerated Lagrangian-type mortar method is devised and followed by a proof of the optimality.
Numerical procedures and discrete equations
To perform a composite solid analysis using the proposed method, typical inclusions of realistic morphology such as reinforced particles or fibers are idealized. For example in microscopic analysis, the realistic morphology can be constructed using several visualization techniques such as atomic force microscopy (AFM), X-ray tomography and serial sectioning [53], [54]. Those inclusions can be digitized and segmented into multiple regions (sets of pixels). The segmentation of these regions in
Numerical examples
In this section, we analyze three simplified-particle models in two-dimension to study the performance of the proposed method in the composite solid problems. Unless otherwise specified, the following conditions are considered: (1) The weight function is chosen to be the cubic B-spline kernel function with normalized support size equal to 1.6 for the construction of meshfree shape functions. (2) A six-point Gauss quadrature rule is used in each integration cell for all examples. (3) The
Conclusions
In this paper, we proposed and analyzed an immersed meshfree method for the elasticity of composite solid problems. This method introduces a new discretization based on the naturally conforming property of meshfree method to approximate the overlapping sub-domains in the composite solid analyses. The method is regarded as a non-conforming method which can be related to the Lagrangian-type mortar method in treating the material discontinuity across the interface. In contrast to the existing
Acknowledgements
The authors would like to thank Dr. John O. Hallquist of LSTC for his support to this research. The support from the Boeing Company is also gratefully acknowledged.
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