Improving the Condition Number of Finite Element Stiffness Matrices via Inverted Elements

2011 ◽  
Author(s):  
Josh Danczyk ◽  
Krishnan Suresh
Keyword(s):  
Finite Element ◽  
Lower Bound ◽  
Condition Number ◽  
Poor Quality ◽  
Iterative Solvers ◽  
Mesh Quality ◽  
Mathematical Properties ◽  
Stiffness Matrices ◽  
Mesh Improvement

In the finite element method, poor quality elements typically increase the condition number of the underlying stiffness matrix, thereby potentially: (1) degrading the computed solution, and (2) slowing the convergence of iterative solvers. Current mesh improvement strategies rely on node movement and edge-flipping to alleviate these problems. However, these methods cannot guarantee a lower-bound on mesh quality, especially in 3-D. In this paper we propose the concept, and use, of inverted elements to improve mesh quality and condition number. Inverted elements are standard finite elements, but with negative Jacobian. After establishing the mathematical properties of these elements we show how they can be used to dramatically improve the quality of a mesh through the use of an ‘element cover’. Further, we show that a lower-bound on the mesh quality can be easily achieved, as supported by numerical experiments and case-studies.

A COST/BENEFIT ANALYSIS OF SIMPLICIAL MESH IMPROVEMENT TECHNIQUES AS MEASURED BY SOLUTION EFFICIENCY

2000 ◽  
Vol 10 (04) ◽  
pp. 361-382 ◽  
Author(s):  
LORI A. FREITAG ◽  
CARL OLLIVIER-GOOCH
Keyword(s):  
Finite Element ◽  
Cost Benefit Analysis ◽  
Cost Benefit ◽  
Poor Quality ◽  
Mesh Improvement ◽  
Simplicial Mesh ◽  
Solution Efficiency ◽  
The Cost ◽  
The Relationship

The quality of unstructured meshes has long been known to affect both the efficiency and the accuracy of the numerical solution of application problems. Mesh quality can often be improved through the use of algorithms based on local reconnection schemes, node smoothing, and adaptive refinement or coarsening. These methods typically incur a significant cost, and in this paper, we provide an analysis of the tradeoffs associated with the cost of mesh improvement in terms of solution efficiency. We first consider simple finite element applications and show the effect of increasing the number of poor quality elements in the mesh and decreasing their quality on the solution time of a number of different solvers. These simple application problems are theoretically well-understood, and we show the relationship between the quality of the mesh and the eigenvalue spectrum of the resulting linear system. We then consider realistic finite element and finite volume application problems, and show that the cost of mesh improvement is significantly less than the cost of solving the problem on a poorer quality mesh.

A Framework for Finite Element Mesh Quality Improvement and Visualization in Orthopaedic Biomechanics

2009 ◽  
Author(s):  
Kiran H. Shivanna ◽  
Srinivas C. Tadepalli ◽  
Vincent A. Magnotta ◽  
Nicole M. Grosland
Keyword(s):  
Finite Element ◽  
Quality Improvement ◽  
Biological Processes ◽  
Mesh Quality ◽  
The Finite Element Method ◽  
Element Mesh ◽  
Invaluable Tool ◽  
Finite Element Meshing ◽  
Mesh Quality Improvement

The finite element method (FEM) is an invaluable tool in the numerical simulation of biological processes. FEM entails discretization of the structure of interest into elements. This discretization process is termed finite element meshing. The validity of the solution obtained is highly dependent on the quality of the mesh used. Mesh quality can decrease with increased complexity of the structure of interest, as is often evident when meshing biologic structures. This necessitated the development/implementation of generalized mesh quality improvement algorithms.

Evaluation of Hexahedral Mesh Quality for Finite Element Method in Electromagnetics

2010 ◽  
Vol 670 ◽  
pp. 318-324 ◽  
Author(s):  
Y. Motooka ◽  
So Noguchi ◽  
H. Igarashi
Keyword(s):  
Finite Element Method ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Computation Time ◽  
Poor Quality ◽  
Mesh Quality ◽  
Hexahedral Mesh ◽  
High Quality ◽  
Element Analysis ◽  
Mesh Generator

We have previously proposed an automatic hexahedral mesh generator. It is necessary to understand about the quality and characteristic of the generated mesh to perform hexahedral edge finite element analysis in electromagnetic. Therefore, we have compared high-quality meshes with poor-quality meshes, and investigated about the factors that affect the accuracy and the computation time. In addition, we investigated about the effect of the templates used in the proposed method. We will conclusively apply the result to improving the automatic hexahedral mesh generator.

Application of an a priori Jacobian-Based Error Estimation Metric to the Accuracy Assessment of 3D Finite Element Simulations

2018 ◽  
Author(s):  
Robert B. Rainsberger ◽  
Jeffrey T. Fong ◽  
Pedro V. Marcal
Keyword(s):  
Finite Element ◽  
Standard Error ◽  
Jacobian Matrix ◽  
Accuracy Assessment ◽  
A Priori ◽  
Element Model ◽  
Finite Element Simulations ◽  
Mesh Quality ◽  
Element Mesh

The determinant of the Jacobian matrix is frequently used in the Finite Element Method as a measure of mesh quality. A new metric is defined, called the Standard Error, based on the distribution of the determinants of the Jacobian matrices of all elements of a finite element mesh. Where the Jacobian norm can be used to compare the quality of one element to another of the same type, the Standard Error compares the mesh quality of different versions of a finite element model where each version uses a different element type. To motivate this new Standard Error, we investigate the geometric meaning of the Jacobian norm on 3D Finite Elements. This mesh quality metric is applied to 8, 20, and 27 node hexahedra, 6 and 15 node prisms, 4 and 10 node tetrahedra, 5 and 13 node pyramid, and 3, 4, 6, 8, and 9 node shell elements. The shape functions for these 14 element types, or more precisely their first partial derivatives, are used to construct the Jacobian Matrix. The matrix is normalized to compensate for size. The determinant of the Jacobian is calculated at Gaussian points within each element. Statistics are gathered to form the Standard Error of the mesh. To illustrate the applicability of this a priori metric, we present two simple example problems having exact answers, and two industry-type problems, a pipe elbow with a crack and a magnetic resonance imaging (MRI) birdcage RF coil resonance, both having no analytical solution. Significance and limitations of using this a priori metric to assess the accuracy of finite element simulations of different mesh designs are presented and discussed.

Superior Mesh Quality With Automatic Node Mergers

1995 ◽  
Author(s):  
Ron S. Gutfinger
Keyword(s):  
Marked Improvement ◽  
Initial Boundary ◽  
Element Model ◽  
Poor Quality ◽  
Mesh Quality ◽  
Boundary Node ◽  
High Quality ◽  
A New Technique ◽  
Mesh Quality Improvement

Abstract Automatic surface meshers have been designed with the intent to eliminate the manual effort involved in creating a finite element model. Naturally, the quality of the mesh must be such that the solver will yield accurate results. For geometries having boundaries with short edges relative to the rest of the geometry, the existing meshers yield poor quality meshes. Hence, the stress analyst must manually attain reasonable mesh quality. This manual effort is very tedious and time consuming. Mesh quality improvement techniques such as smoothing, fail to improve the mesh quality. In this work, a new technique is introduced, which improves the initial boundary node loop quality, and in turn yields high quality meshes. Examples show marked improvement in quality, and the meshes are comprised of fewer elements.

SINK INSERTION FOR MESH IMPROVEMENT

2002 ◽  
Vol 13 (02) ◽  
pp. 223-242 ◽  
Author(s):  
HERBERT EDELSBRUNNER ◽  
DAMRONG GUOY
Keyword(s):  
New Technique ◽  
Mesh Quality ◽  
Insertion Technique ◽  
Delaunay Triangulations ◽  
Running Time ◽  
Mesh Improvement ◽  
Numerical Robustness ◽  
A New Technique

We propose sink insertion as a new technique to improve the mesh quality of Delaunay triangulations. We compare it with the conventional circumcenter insertion technique under three scheduling regimes: incremental, in blocks, and in parallel. Justification for sink insertion is given in terms of mesh quality, numerical robustness, running time, and ease of parallelization.

scholarly journals Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Carola Kruse ◽  
Vincent Darrigrand ◽  
Nicolas Tardieu ◽  
Mario Arioli ◽  
Ulrich Rüde
Keyword(s):  
Finite Element ◽  
Linear System ◽  
Iterative Method ◽  
Condition Number ◽  
Degrees Of Freedom ◽  
Mesh Size ◽  
Element Model ◽  
Structural Mechanics ◽  
Iterative Solvers ◽  
Test Problems

AbstractKinematic relationships between degrees of freedom, also named multi-point constraints, are frequently used in structural mechanics. In this paper, the Craig variant of the Golub-Kahan bidiagonalization algorithm is used as an iterative method to solve the arising linear system with a saddle point structure. The condition number of the preconditioned operator is shown to be close to unity and independent of the mesh size. This property is proved theoretically and illustrated on a sequence of test problems of increasing complexity, including concrete structures enforced with pretension cables and the coupled finite element model of a reactor containment building. The Golub-Kahan algorithm converges in only a small number of steps for all considered test problems and discretization sizes. Furthermore, it is robust in practical cases that are otherwise considered to be difficult for iterative solvers.

An experience of performing spirometry by trained general practitioners

2020 ◽  
pp. 34-36
Author(s):  
M. A. Pokhaznikova ◽  
E. A. Andreeva ◽  
O. Yu. Kuznetsova
Keyword(s):  
General Practitioners ◽  
Quality Criteria ◽  
Poor Quality ◽  
Male Gender ◽  
Study Research ◽  
The Poor ◽  
Quality Of Research ◽  
Forced Expiratory Maneuver ◽  
Maximal Effort

The article discusses the experience of teaching and conducting spirometry of general practitioners as part of the RESPECT study (RESearch on the PrEvalence and the diagnosis of COPD and its Tobacco-related aetiology). A total of 33 trained in spirometry general practitioners performed a study of 3119 patients. Quality criteria met 84.1% of spirometric studies. The analysis of the most common mistakes made by doctors during the forced expiratory maneuver is included. The most frequent errors were expiration exhalation of less than 6s (54%), non-maximal effort throughout the test and lack of reproducibility (11.3%). Independent predictors of poor spirogram quality were male gender, obstruction (FEV1 /FVC<0.7), and the center where the study was performed. The number of good-quality spirograms ranged from 96.1% (95% CI 83.2–110.4) to 59.8% (95% CI 49.6–71.4) depending on the center. Subsequently, an analysis of the reasons behind the poor quality of research in individual centers was conducted and the identified shortcomings were eliminated. The poor quality of the spirograms was associated either with the errors of the doctors who undertook the study or with the technical malfunctions of the spirometer.

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