We propose a 3D mesh curving method that converts a straight-sided mesh to an optimal-quality curved high-order mesh that interpolates a CAD boundary representation. The main application of this method is the generation of discrete approximations of curved domains that are valid for simulation analysis with unstructured high-order methods. We devise the method as follows. First, the boundary of a straight-sided high-order mesh is curved to match the curves and surfaces of a CAD model. Second, the method minimizes the volume mesh distortion with respect to the coordinates of the inner nodes and the parametric coordinates of the curve and surface nodes. The proposed minimization features untangling capabilities and therefore, it repairs the invalid elements that may arise from the initial curving step. Compared with other mesh curving methods, the only goal of the proposed residual system is to minimize the volume mesh distortion. Furthermore, it is less constrained since the boundary nodes are free to slide on the CAD curves and surfaces. Hence, the proposed method is well suited to generate curved high-order meshes of optimal quality from CAD models that contain thin parts or high-curvature entities. To illustrate these capabilities, we generate several curved high-order meshes from CAD models with the implementation detailed in this work. Specifically, we detail a node-by-node non-linear iterative solver that minimizes the proposed objective function in a block Gauss-Seidel manner.Peer ReviewedPostprint (author's final draft

We present a novel methodology to generate curved high-order meshes featuring optimal mesh quality and geometric accuracy. The proposed technique combines a distortion measure and a geometric Full-size image (<1 K)-disparity measure into a single objective function. While the element distortion term takes into account the mesh quality, the Full-size image (<1 K)-disparity term takes into account the geometric error introduced by the mesh approximation to the target geometry. The proposed technique has several advantages. First, we are not restricted to interpolative meshes and therefore, the resulting mesh approximates the target domain in a non-interpolative way, further increasing the geometric accuracy. Second, we are able to generate a series of meshes that converge to the actual geometry with expected rate while obtaining high-quality elements. Third, we show that the proposed technique is robust enough to handle real-case geometries that contain gaps between adjacent entities.This research was partially supported by the Spanish Ministerio de Economía y Competitividad under grand contract\ud CTM2014-55014-C3-3-R, and by the Government of Catalonia under grand contract 2014-SGR-1471. The work of the last author was supported by the European Commission through the Marie Sklodowska-Curie Actions\ud (HiPerMeGaFlows project).Peer ReviewedPostprint (published version

We formulate a mesh morphing technique as mesh distortion minimization problem constrained to weakly satisfy the imposed displacement of the boundary nodes. The method is devised to penalize the appearance of inverted elements during the optimization process. Accordingly, we have not equipped the method with untangling capabilities. To solve the constrained minimization problem, we apply the augmented Lagrangian technique to incorporate the boundary condition in the objective function using the Lagrangian multipliers and a penalty parameter. We have applied the proposed formulation to mesh moving and mesh curving problems. The results show that the method has the ability to deal with large displacements for 2D and 3D meshes with non-uniform sizing, and mesh curving of highly stretched 2D high-order meshes.This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No 715546. The work of the corresponding author has been partially supported by the Spanish Ministerio de Economía y Competitividad under the personal grant agreement RYC- 2015-01633. The work of the third author has been supported by the Spanish Ministerio de Economía y\ud Competitividad under grant agreement CTM2014-55014-C3-3-R.Peer ReviewedPostprint (published version

Two of the most successful methods to generate unstructured hexahedral meshes are the grid-based methods and the advancing front methods. On the one hand, the grid-based methods generate high-quality hexahedra in the inner part of the domain using an inside-outside approach. On the other hand, advancing front methods generate highquality hexahedra near the boundary using an outside-inside approach. To combine the advantages of both methodologies, we extend the receding front method: an inside-outside mesh generation approach by means of a reversed advancing front. We apply this approach to generate unstructured hexahedral meshes of exterior domains. To reproduce the shape of the boundaries, we first pre-compute the mesh fronts by combining two solutions of the Eikonal equation on a tetrahedral reference mesh. Then, to generate high-quality elements, we expand the quadrilateral surface mesh of the inner body towards the unmeshed external boundary using the pre-computed fronts as a guide.

We present a high-order mesh curving method where the mesh boundary is enforced to match a target virtual geometry. Our method has the unique capability to allow curved elements to span and slide on top of several CAD entities during the mesh curving process. The main advantage is that small angles or small patches of the CAD model do not compromise the topology, quality and size of the boundary elements. We associate each high-order boundary node to a unique group of either curves (virtual wires) or surfaces (virtual shell). Then, we deform the volume elements to accommodate the boundary curvature, while the boundary condition is enforced with a penalty method. At each iteration of the penalty method, the boundary condition is updated by projecting the boundary interpolative nodes of the previous iteration on top of the corresponding virtual entities. The method is suitable to curve meshes featuring non-uniform isotropic and highly stretched elements while matching a given virtual geometry.

In this work, we present a simultaneous untangling and smoothing technique for quadrilateral and hexahedral meshes. The algorithm iteratively improves a quadrilateral or hexahedral mesh by minimizing an objective function defined in terms of a regularized algebraic distortion measure of the elements. We propose several techniques to improve the robustness and the computational efficiency of the optimization algorithm. In addition, we have adopted an object-oriented paradigm to create a common framework to smooth meshes composed by any type of elements, and using different minimization techniques. Finally, we present several examples to show that the proposed technique obtains valid meshes composed by high-quality quadrilaterals and hexahedra, even when the initial meshes contain a large number of tangled elements.

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We develop a high-order hybridizable discontinuous Galerkin (HDG) formulation to solve the immiscible and incompressible two-phase flow problem in a heterogeneous porous media. The HDG method is locally conservative, has fewer degrees of freedom than other discontinuous Galerkin methods due to the hybridization procedure, provides built-in stabilization for arbitrary polynomial degrees and, if the error of the temporal discretization is low enough, the pressure, the saturation and their fluxes converge with order P + 1 in L 2 -norm, being P the polynomial degree. In addition, an element-wise post-process can be applied to obtain a convergence rate of P + 2 in L 2 -norm for the scalar variables. All of these advantages make the HDG method suitable for solving multiphase flow trough porous media. We show numerical evidence of the convergences rates. Finally, to assess the capabilities of the proposed formulation, we apply it to several cases of water-flooding technique for oil recovery. KEYWORDSTwo-phase flow; immiscible; incompressible; hybridizable discontinuous Galerkin method; diagonally implicit Runge-Kutta method; differential algebraic equations.