High-order Mesh Research Articles

The multi-level [formula omitted]-method for three-dimensional problems: Dynamically changing high-order mesh refinement with arbitrary hanging nodes

One main challenge of the hp-version of the finite element method is the high implementational complexity of the method resulting from the added need of handling hanging nodes appropriately. The multi-level hp-formulation–recently introduced for two-dimensional applications–aims at alleviating these difficulties without compromising the approximation quality. This is achieved by changing from the conventional refine-by-replacement approach to a refine-by-superposition idea. The current work shows that the multi-level hp-approach can be extended naturally to three-dimensional refinement without increasing the complexity of the rule set ensuring linear independence and compatibility of the shape functions. In this way, a three-dimensional hp-refinement scheme is formulated, in which hanging nodes are avoided by definition. This ease of complexity allows for a highly flexible discretization kernel featuring arbitrary irregular meshes and a continuous refinement and coarsening throughout the simulation runtime. Different numerical examples demonstrate that–even in the presence of singularities–this novel refinement scheme yields exponential convergence with respect to both, the number of unknowns and the computational time. It is further shown that the refinement scheme is able to capture complex solution features that demand for three-dimensional refinement patterns. The dynamic discretization properties of the approach are demonstrated by continuously refining and coarsening the mesh during the simulation runtime to keep the refinement zone local to a moving singularity. Finally, it is shown that the high approximation power of the multi-level hp-scheme also carries over to curved geometries common in engineering practice without a significant detrimental effect on the conditioning of the stiffness matrix.

A unified approach for a posteriori high-order curved mesh generation using solid mechanics

The paper presents a unified approach for the a posteriori generation of arbitrary high-order curvilinear meshes via a solid mechanics analogy. The approach encompasses a variety of methodologies, ranging from the popular incremental linear elastic approach to very sophisticated non-linear elasticity. In addition, an intermediate consistent incrementally linearised approach is also presented and applied for the first time in this context. Utilising a consistent derivation from energy principles, a theoretical comparison of the various approaches is presented which enables a detailed discussion regarding the material characterisation (calibration) employed for the different solid mechanics formulations. Five independent quality measures are proposed and their relations with existing quality indicators, used in the context of a posteriori mesh generation, are discussed. Finally, a comprehensive range of numerical examples, both in two and three dimensions, including challenging geometries of interest to the solids, fluids and electromagnetics communities, are shown in order to illustrate and thoroughly compare the performance of the different methodologies. This comparison considers the influence of material parameters and number of load increments on the quality of the generated high-order mesh, overall computational cost and, crucially, the approximation properties of the resulting mesh when considering an isoparametric finite element formulation.

High-Order Mesh Generation for Discontinuous Galerkin Methods Based on Elastic Deformation

AbstractIn this paper, a high-order curved mesh generation method for Discontinuous Galerkin methods is introduced. First, a regular mesh is generated. Second, the solid surface is re-constructed using cubic polynomial. Third, the elastic governing equations are solved using high-order finite element method to provide a fully or partly curved grid. Numerical tests indicate that the intersection between element boundaries can be avoided by carefully defining the elasticity modulus.

Implementation of Convergence in Adaptive Global Digital Image Correlation

In FE based global digital image correlation (DIC) a finite element mesh is used to describe the deformation of the region of interest (ROI). However, the identification of an optimal mesh is a difficult problem and is often obtained by using “mechanical” pre-knowledge of the solution. In Finite Element Analysis (FEA) an optimal mesh can be found without any pre-knowledge of the solution by using mesh adaptivity, where an initial (non optimal) mesh is refined until the optimal solution is obtained. Refinement of the mesh can be based on error and/or convergence estimators. Despite the fundamental differences between FEA and DIC, in the present article the convergence procedure is successfully used in a recently published global FE based DIC method. In the used global DIC method elements can receive higher order shape functions, also known as p-elements. Using the aforementioned algorithm, also called p-DIC, refinement to a non-uniform higher order mesh is possible. Using the non-uniform mesh, an optimal mesh can be obtained for each section of the ROI. The presented study shows that a convergence scheme can be used to automatically control the mesh refinement in a global DIC approach. The convergence boundary, in percentage, is a more intuitive boundary than the absolute error boundary used in the original p-DIC approach. The procedure is validated using numerical examples and the robustness to experimental variables is investigated. Finally, the complete procedure is tested against a wide range of practical examples.

Optimizing the geometrical accuracy of curvilinear meshes

This paper presents a method to generate valid high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a linear mesh, that is subsequently curved without taking care of the validity of the high order elements. An optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh. Standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two-dimensional meshes, but they prove computationally too costly for optimization purposes. A fast estimate of the geometrical accuracy, based on Taylor expansions of the curves, is introduced. An unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes, as measured by the standard Haudorff distance between the geometrical model and the mesh. Several examples illustrate the beneficial impact of this method on CFD solutions, with a particular role of the enhanced mesh boundary smoothness.

A Variational Framework for High-order Mesh Generation

The generation of sufficiently high quality unstructured high-order meshes remains a significant obstacle in the adoption of high-order methods. However, there is little consensus on which approach is the most robust, fastest and produces the ‘best’ meshes. We aim to provide a route to investigate this question, by examining popular high-order mesh generation methods in the context of an efficient variational framework for the generation of curvilinear meshes. By considering previous works in a variational form, we are able to compare their characteristics and study their robustness. Alongside a description of the theory and practical implementation details, including an efficient multi-threading parallelisation strategy, we demonstrate the effectiveness of the framework, showing how it can be used for both mesh quality optimisation and untangling of invalid meshes.

Generation of Curved High-order Meshes with Optimal Quality and Geometric Accuracy

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 ▪-disparity measure into a single objective function. While the element distortion term takes into account the mesh quality, the ▪-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.

High-order mesh curving by distortion minimization with boundary nodes free to slide on a 3D CAD representation

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.

An Improved XFEM With Multiple High-Order Enrichment Functions and Low-Order Meshing Elements for Field Analysis of Electromagnetic Devices With Multiple Nearby Geometrical Interfaces

This paper proposes an improved extended finite element method for modeling electromagnetic devices with multiple nearby geometrical interfaces. In regions near these interfaces, the magnetic vector potential approximation is enriched by incorporating multiple derivative discontinuous fields based on the partition of unity method such that the interfaces can be represented independent of the mesh. The support of a node or an element can be cut by several interfaces. This method results in the high accuracy in the approximation field and derivative field. Numerical examples applied to the iron core in 1-D eddy current field involving level set-based parts, error analysis, and electromagnetic field computations are provided to demonstrate the utility of the proposed approach.

Multi-level hp-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes

The implementation of $$hp$$hp-adaptivity is challenging as hanging nodes, edges, and faces have to be constrained to ensure compatibility of the shape functions. For this reason, most $$hp$$hp-code frameworks restrict themselves to $$1$$1-irregular meshes to ease the implementational effort. This work alleviates these difficulties by introducing a new formulation for high-order mesh adaptivity that provides full local $$hp$$hp-refinement capabilities at a comparably small implementational effort. Its main idea is the extension of the $$hp$$hp-$$d$$d-method such that it allows for high-order overlay meshes yielding a hierarchical, multi-level $$hp$$hp-formulation of the Finite Element Method. This concept enables intuitive refinement and coarsening procedures, while linear independence and compatibility of the shape functions are guaranteed by construction. The proposed method is demonstrated to achieve exponential rates of convergence--both in terms of degrees of freedom and in run-time--for problems with non-smooth solutions. Furthermore, the scheme is used alongside the Finite Cell Method to simulate the heat flow around moving objects on a non-conforming background mesh and is combined with an energy-based refinement indicator for automatic $$hp$$hp-adaptivity.

Defining an 2-disparity Measure to Check and Improve the Geometric Accuracy of Non-interpolating Curved High-order Meshes

We define an ▪2-disparity measure between curved high-order meshes and parameterized manifolds in terms of an ▪2 norm. The main application of the proposed definition is to measure and improve the distance between a curved high-order mesh and a target parameterized curve or surface. The approach allows considering meshes with the nodes on top of the curve or surface (interpolative), or floating freely in the physical space (non-interpolative). To compute the disparity measure, the average of the squared point-wise differences is minimized in terms of the nodal coordinates of an auxiliary parametric high-order mesh. To improve the accuracy of approximating the target manifold with a non-interpolating curved high-order mesh, we minimize the square of the disparity measure expressed both in terms of the nodal coordinates of the physical and parametric curved high-order meshes. The proposed objective functions are continuously differentiable and thus, we are able to use minimization algorithms that require the first or the second derivatives of the objective function. Finally, we present several examples that show that the proposed methodology generates high-order approximations of the target manifold with optimal convergence rates for the geometric accuracy even when non-uniform parameterizations of the manifolds are prescribed. Accordingly, we can generate coarse curved high-order meshes significantly more accurate than finer low-order meshes that feature the same resolution.

Distortion and quality measures for validating and generating high-order tetrahedral meshes

A procedure to quantify the distortion (quality) of a high-order mesh composed of curved tetrahedral elements is presented. The proposed technique has two main applications. First, it can be used to check the validity and quality of a high-order tetrahedral mesh. Second, it allows the generation of curved meshes composed of valid and high-quality high-order tetrahedral elements. To this end, we describe a method to smooth and untangle high-order tetrahedral meshes simultaneously by minimizing the proposed mesh distortion. Moreover, we present a \(p\)-continuation procedure to improve the initial configuration of a high-order mesh for the optimization process. Finally, we present several results to illustrate the two main applications of the proposed technique.

Optimizing the Geometrical Accuracy of 2D Curvilinear Meshes

This paper presents a method to generate valid 2D high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a straight sided mesh. High order points are initially snapped to the real geometry without taking care of the validity of the high order elements. An optimization procedure that both allow to untangle invalid elements and to optimize the geometrical accuracy of the mesh is presented. An area based distance is proposed to compute the geometrical discrepancy between the mesh and its underlying geometry. This area based distance is shown to be strongly related to standard distance measures such as the Hausdorff distance, while being largely faster to compute.The new distance is minimized through an unconstrained optimization procedure, allowing significant improvements of the geometrical accuracy of high order meshes. A simple computational example shows that geometrically optimized meshes allow smooth CFD solutions.

Automatic Curvilinear Quality Mesh Generation Driven by Smooth Boundary and Guaranteed Fidelity

The development of robust high-order finite element methods requires the construction of valid high-order meshes for complex geometries without user intervention. This paper presents a novel approach for automatically generating a high-order mesh with two main features: first, the boundary of the mesh is globally smooth; second, the mesh boundary satisfies a required fidelity tolerance. Invalid elements are eliminated. Example meshes demonstrate the features of the algorithm.

A surface mesh smoothing and untangling method independent of the CAD parameterization

A method to optimize triangular and quadrilateral meshes on parameterized surfaces is proposed. The optimization procedure relocates the nodes on the surface to improve the quality (smooth) and ensures that the elements are not inverted (untangle). We detail how to express any measure for planar elements in terms of the parametric coordinates of the nodes. The extended measures can be used to check the quality and validity of a surface mesh. Then, we detail how to optimize any Jacobian-based distortion measure to obtain smoothed and untangled meshes with the nodes on the surface. We prove that this method is independent of the surface parameterization. Thus, it can optimize meshes on CAD surfaces defined by low-quality parameterizations. The examples show that the method can optimize meshes composed by a large number of inverted elements. Finally, the method can be extended to obtain high-order meshes with the nodes on the CAD surfaces.

Adaptation based on interpolation errors for high order mesh refinement methods applied to conservation laws

Adaptive mesh refinement is nowadays a widely used tool in the numerical solution of hyperbolic partial differential equations. The algorithm is based on the numerical approximation of the solution of the equations on a hierarchical set of meshes with different resolutions. Among the different parts that compose an adaptive mesh refinement algorithm, the decision of which level of resolution is adequate for each part of the domain, i.e., the design of a refinement criterion, is crucial for the performance of the algorithm. In this work we analyze a refinement strategy based on interpolation errors, as a building block of a high order adaptive mesh refinement algorithm. We show that this technique, inspired by multiresolution algorithms, is competitive, in terms of the balance between accuracy of the solutions and computational time, against adaptation methods based on a gradient sensor.

Introducing the target-matrix paradigm for mesh optimization via node-movement

A general-purpose algorithm for mesh optimization via node-movement, known as the Target-Matrix Paradigm, is introduced. The algorithm is general purpose in that it can be applied to a wide variety of mesh and element types, and to various commonly recurring mesh optimization problems such as shape improvement, and to more unusual problems like boundary-layer preservation with sliver removal, high-order mesh improvement, and edge-length equalization. The algorithm can be considered to be a direct optimization method in which weights are automatically constructed to enable definitions of application-specific mesh quality. The high-level concepts of the paradigm have been implemented in the Mesquite mesh improvement library, along with a number of concrete algorithms that address mesh quality issues such as those shown in the examples of the present paper.

A Locally Corrected Nyström Formulation for the Magnetostatic Volume Integral Equation

A locally corrected Nystrom discretization of the magnetostatic volume integral equation is derived for the analysis of magnetic materials. The integral equation formulation incorporates both higher-order meshes and higher-order basis functions. A set of arbitrary order, hexahedral basis functions are presented. The formulation is applied to a set of canonical problems as well as TEAM Workshop problem number 13. Error convergence with respect to basis function order, mesh density, and mesh order is investigated, and results corroborate the formulation.

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex ge- ometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order techniques in both academia (where the use of low-order schemes remains widespread) and industry (where the use of low-order schemes is ubiquitous). In this short review, issues that have hitherto pre- vented the use of high-order methods amongst a non-specialist community are identified, and cur- rent efforts to overcome these issues are discussed. Attention is focused on four areas, namely the generation of unstructured high-order meshes, the development of simple and efficient time integration schemes, the development of robust and accurate shock capturing algorithms, and fi- nally the development of high-order methods that are intuitive and simple to implement. With regards to this final area, particular attention is focused on the recently proposed flux reconstruc- tion approach, which allows various well known high-order schemes (such as nodal discontinuous Galerkin methods and spectral difference methods) to be cast within a single unifying framework. It should be noted that due to the experience of the authors the review is directed somewhat to- wards aerodynamic applications and compressible flow. However, many of the discussions have a wider applicability. Moreover, the tone of the review is intended to be generally accessible, such that an extended scientific community can gain insight into factors currently pacing the adoption of unstructured high-order methods.

Using coordinate transformation of Navier–Stokes equations to solve flow in multiple helical geometries

Recent research on small amplitude helical pipes for use as bypass grafts and arterio-venous shunts, suggests that mixing may help prevent occlusion by thrombosis. It is proposed here that joining together two helical geometries, of different helical radii, will enhance mixing, with only a small increase in pressure loss. To determine the velocity field, a coordinate transformation of the Navier–Stokes equations is used, which is then solved using a 2-D high-order mesh combined with a Fourier decomposition in the periodic direction. The results show that the velocity fields in each component geometry differ strongly from the corresponding solution for a single helical geometry. The results suggest that, although the mixing behaviour will be weaker than an idealised prediction indicates, it will be improved from that generated in a single helical geometry.