A Virtual Element Approach for Non-Linear Problems on Arbitrary Distorted 8-Node Bricks
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The isoparametric Finite Element Method is a common choice to solve non-linear problems in engineering and technology. However, it has certain limitations regarding mesh quality and the presence of ill-conditioned elements. It is typical to manipulate the mesh to achieve its well-posedness; however, the associated costs, in terms of time and resources, can be significant. A recent alternative is the use of the Virtual Element Method (VEM) a generalization of the Finite Element Method, which allows using very general polygonal and polyhedral meshes. The main drawback of this approach is its time-consuming nature, especially in three-dimensional applications. In our work, we aim to introduce a new formulation for the VEM in three dimensions on general hexahedrons. This new space is defined as the set of functions that are harmonic and continuous on the boundary such that the functions are bilinear on the faces, as in first order isoparametric FEM. This idea differs from the classical approach where the functions must be polynomial only on the edges, while on the faces, they are virtual, limiting the applicability of the method to polyhedral. In contrast, in our formulation the shape functions on the faces are explicitly known. The new approach leads to a straightforward computation of the nabla projector, reducing the computational cost. Moreover, the space and degrees of freedom on the faces are the same as those of the classical Lagrangian FEM, making it possible to easily construct a mixed method in which VE and FE are present in the same mesh. The proposed method has been tested on nonlinear problems like the Poisson equation with nonlinear diffusion coefficient and nonlinear elastic problems. The results are encouraging, especially on highly distorted meshes.