Stability of mixed overlapping elements in incompressible analysis
Introduction
Finite element analysis (FEA) is an efficient numerical technique that has been widely used in many scientific fields including complex analysis of solid, fluid, or multi-physics systems. Due to its robustness and efficiency in solving boundary value problems, FEA has undergone extensive developments over the decades [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. Even though FEA is an effective tool, it still has a few challenges, one of which is the cumbersome meshing procedure. The mesh refinement level should be carefully managed by engineers to adjust a balance between the computational time and the desired solution accuracy. More refinement should be considered for the areas with concentrated stress gradients or singular geometries [7]. In the case of large deformation or crack analyses, distortion of the mesh may occur, which may decrease the accuracy of the solution [13], [14], [15], [16]. Furthermore, it is often challenging to build a conformal mesh when an object is desired to be constructed as hexahedral mesh, especially if the object has a complex geometry, such as part-to-part interfaces of industrial components or biological molecular surfaces [17], [18], [19]. This meshing difficulty sometimes confines the discretization to the use of triangular or tetrahedral meshes, whose meshing process is rather straightforward at the expense of accuracy. Therefore, the meshing procedure requires the labor-intensive judgment of a skilled engineer, which might make the entire process inefficient.
Alternative numerical techniques have been suggested to avoid such difficulties in the meshing process. By developing an element-free Galerkin method, Belytschko et al. opened up new possibilities for meshless methods [20]. Liu et al. expanded the field of meshless methods by introducing a smoothed particle hydrodynamics (SPH) approach [21]. The technique was originally developed for solving astrophysical problem [22]; however, the method has been successfully implemented to solve engineering problems including fluid flow, fracture, fluid–solid interaction, and heat conduction [23], [24], [25], [26], [27]. De et al. proposed a method of finite spheres focusing on performing numerical integration without a mesh and treating Dirichlet boundary conditions [28], [29], [30]. They successfully implemented the method mainly to solve elasticity problems such as three-dimensional static analysis or transient wave propagation problems as an alternative to traditional FEA [31], [32]. Another approach to overcoming meshing difficulty is enhancing the traditional finite elements using an enrichment function. Bathe and Almeida adopted an enrichment function to capture the cubic behavior of ovalization in pipe problems [33], [34], [35]. Bathe and Chaudhary expanded their work to beam problems with the warping effect [36]. Moës and Belytschko proposed an extended finite element method (XFEM) [37]. The method was developed based on the partition of unity and the generalized finite element method and was successfully adopted in capturing the cracks or material discontinuity [38], [39].
More recently, Kim and Bathe proposed the finite elements enriched by interpolation covers [40]. Their work contributed to increasing the convergence rate based on the low-order triangular or tetrahedral mesh. Zhang and Bathe expanded their work by presenting an overlapping element [41], [42], [43], [44], [45], [46], [47]. The overlapping element was established based on the finite element enriched by interpolation cover and additional theories from the method of finite spheres. The term “overlap” was first introduced in their work on overlapping element [46], which aimed to address the difficulties of meshing in traditional finite elements. By designing a newly developed overlapping elements, which actually ‘overlap’ over the analyzed region, the authors were able to achieve efficiency by combining the overlapping element with the traditional finite elements. The original approach utilized spherical and brick-shaped overlapping element, but the method was later improved with the introduction of triangular-based overlapping element that maintained the characteristics of insensitivity to mesh distortion while also enabling more efficient numerical integration [41]. The technique was further enhanced by incorporating virtual nodes, which are used to estimate the local field and then globally interpolated, resulting in reduced bandwidth of resulting matrices [42]. As a result of these improvements, the overlapping element offers both efficiency and robustness with more efficient numerical integration. This novel meshing scheme has been the subject of continued research to explore its potential [43], [44], [48], [49], [50]. In contrast to traditional low-order triangular or tetrahedral finite elements, which are less accurate and exhibit overly stiff behavior, the overlapping element provides effective solutions using low-order triangular or tetrahedral mesh. Additionally, the method features the insensitivity to mesh distortion. Due to the benefits of overlapping element, the overlapping element has been successfully implemented in the analysis of structure subject to thermal loads with spatially varying gradients [51].
The core concept of overlapping element shares similarities with the High-Continuity (HC) finite element introduced by Aristodemo [52], in that both methods utilize information from adjacent regions to improve local estimations. Further investigations into HC element by Daniel [53] extended their use to transient dynamic problem and plate problems. In pursuit of inter-element continuity in displacement and its derivatives, the term ‘overlapping’ was adopted as the integrated element is influenced by adjacent overlapped elements. The HC method proved successful in achieving high continuity using only a limited number of degrees of freedom, offering benefits not only for plate problem where the continuity of derivatives is crucial but also for general two-dimensional plane-stress problems when compared to traditional Q4 or Q6 elements. More recently, the method has been expanded to general three-dimensional problems through the adoption of B-spline or NURBS interpolation method [54]. While the HC element focuses on maintaining inter-element continuity in strain and stress with reduced degrees of freedom, the overlapping element prioritizes alleviating the laborious meshing process through the benefits of insensitivity to mesh distortion, numerical efficiency, and robustness.
The displacement–pressure mixed FEA is another important research topic that has been studied in incompressible analysis. It is well known that the pure displacement-based finite element may lead to volumetric locking problem when solving for incompressible or near-incompressible media [7]. To address this volumetric locking problem, many researchers devoted to developing mixed finite elements [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65]. Unlike traditional FEA, mixed FEA adopts at least two unknown variables (usually displacement and pressure). The mixed FEA typically has the linear algebraic system of the form: where and are matrices resulting from the governing equations, and are the vectors of unknown variables, and and are external load vectors. In mixed FEA, the stability is an important property in addition to the solvability. The stability of the mixed FEA is ensured by examining whether the matrix in Eq. (1) satisfies the stability condition. Brezzi and Babuška presented the inf–sup (or Brezzi–Babuška) condition as a criterion for ensuring the stability of mixed finite elements [58], [66], [67]. Subsequently, Chapelle and Bathe proposed a numerical procedure for testing inf–sup condition to make the inf–sup test accessible for newly developed mixed finite elements [68]. Unfortunately, owing to the rigorous requirements of this stability condition, the ratio of the number of displacement and pressure unknowns is constrained to a specific level, and this frequently leads to the use of only high-order elements for traditional mixed finite elements which are computationally expensive. It would be ideal if we can achieve a stable mixed formulation using the overlapping element which is based on low-order triangular mesh.
In this study, we implemented the displacement–pressure mixed formulation in the overlapping element. Owing to its enlarged displacement function spaces, the overlapping element can embrace the pressure variables without violating the stability condition. Because the overlapping element is based on the low-order triangular meshes, it is computationally efficient, and the meshing process becomes much simpler. We present the reasonable ratio of displacement–pressure variables that can be used for the mixed overlapping element with a sequence of stability tests and numerical examples. Additionally, we demonstrate that the mixed overlapping element still has the original property of the pure displacement-based overlapping element, which is the insensitivity to mesh distortions.
In Section 2, we briefly introduce the formulation of the displacement field in the original pure displacement-based overlapping element in triangular meshes. The equivalent shape function vector derived from the overlapping elements is also presented. In Section 3, we present the procedure for constructing the matrices for the displacement–pressure mixed overlapping element. In Section 4, we introduce the procedure for the numerical inf–sup stability test that is applicable to mixed overlapping elements. In Section 5, we present numerical examples, including the stability test results, and benchmark problems to highlight the salient features of mixed overlapping elements such that the mixed overlapping element is still insensitive to mesh distortion and accurate in near-incompressible analysis.
This article is based on and expands upon a part of the first author’s Ph.D. thesis work [69] under the supervision of the Department of Mechanical Engineering, Seoul National University.
Section snippets
Displacement approximation of overlapping element
The overlapping element adopts an alternative displacement approximation derived from the theories of the classical FEA [7] and the method of finite spheres [28], [29], [30], [31], [32]. We employed mathematical notations similar to those previously suggested by Huang and Bathe [43].
In the classical FEA, the displacement is approximated by interpolating the nodal unknowns of each element, which is written as where is the number of nodes determined by the shape and order of
Displacement–pressure mixed formulation of overlapping element
The weak form of the governing equation for elasticity is obtained by taking the stationarity of the modified potential (given in Eq. (22)) which results in two separate equations described in Eqs. (23), (24). The modified potential was designed to circumvent the locking phenomenon by weakening the constraint applied on . Here, and are the finite element spaces where
Stability of mixed overlapping elements
For pure displacement-based finite elements, the continuity and ellipticity conditions are the fundamental conditions to guarantee those error bounds, which means that the element exhibits monotonic convergence and a consistent convergence rate on refining meshes [73]. Huang and Bathe have proven in their recent work that the pure displacement-based overlapping element also exhibits monotonic convergence and a consistent convergence rate [44]. In mixed formulation, however, convergence is not
Numerical inf–sup test
We present the numerical inf–sup test results for the proposed mixed overlapping elements. By comparing the inf–sup test results of mixed overlapping elements to those of traditional mixed finite elements, we determine whether each tested mixed overlapping element passes the numerical inf–sup test. The mixed overlapping elements that pass the inf–sup test will be considered stable in an incompressible analysis.
The description for the mechanical system used in the numerical inf–sup tests are
Conclusion
We implemented the displacement–pressure mixed formulation to the overlapping element in pursuit of stable efficient incompressible analysis. By performing a sequence of numerical inf–sup test and numerical examples, we confirmed the stability and effectiveness of mixed overlapping element. We were able to establish a stable mixed formulation of the overlapping element based on low-order triangular mesh with improved computational efficiency, in contrast to the traditional mixed finite
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the Agency For Defense Development Grant funded by the Korean Government (UD220004JD) and Korea Midland Power Co., LTD . (KOMIPO).
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