Barna A. Szabó

The p-Version of the Finite Element Method

Abstract : The finite element method has become the main tool in computational mechanics. The MAKABASE contains about 20,000 references on finite element and 2000 boundary element technology. Recently the new direction in the finite element theory and practice appeared, the p and h-p versions, which utilize high degree of elements. About 3 - 4 dozen references about p and h-p versions are available, all of them related to the elliptic problems. For the survey of todays state of the art, we refer to e.g. This paper addresses the basic problems of the p-version for the parabolic equation with both variables, x and t discreted via p-version. It concentrates on the case when in the time variables only one interval is used. The paper gives the error estimates and presents some numerical aspects. The authors restrict themselves to the basic features of the method. Various generalizations will be presented in forthcoming papers.

Mesh design for the p-version of the finite element method

Abstract When properly designed meshes are used then the performance of the p -extension is very close to the best performance attainable by the finite element method. Proper mesh design depends on the exact solution, however. Because the exact solution is not known a priori, initial mesh design is generally based on certain assumptions concerning the exact solution which must be tested in the post-solution phase to ensure reliability and accuracy of data computed from the finite element solution. In this paper general guidelines are presented for prior design of meshes, and procedures for post-solution testing are described and illustrated by examples.

An inverse approach to determining myocardial material properties

Passive myocardial material properties have been measured previously by subjecting test samples of myocardium to in vitro load-deformation analysis or, in the intact heart, by pressure-volume relationships. A new method for determining passive material properties, described in this paper, couples a p-version finite element model of the heart, a nonlinear optimization algorithm and a dense set of transmural measured strains that could be obtained in the intact heart by magnetic resonance imaging (MRI) radiofrequency tissue tagging. Unknown material parameters for a nonlinear, nonhomogeneous material law are determined by solving an inverse boundary value problem. An objective function relating the least-squares difference of model-predicted and measured strains is minimized with respect to the unknown material parameters using a novel optimization algorithm that utilizes forward finite element solutions to calculate derivatives of model-predicted strains with respect to the material parameters. Test cases incorporating several salient features of the inverse material identification problem for the heart are formulated to test the performance of the inverse algorithm in typical experimental conditions. Known true material parameters can be determined to within a small tolerance and random noise is shown not to affect the stability of the inverse solution appreciably. On the basis of these validation experiments, we conclude that the inverse material identification problem for the heart can be extended to solve for unknown material parameters that describe in vivo myocardial material behavior.

Hierarchic models for laminated plates and shells

The definition, essential properties and formulation of hierarchic models for laminated plates and shells are presented. The hierarchic models satisfy three essential requirements: approximability; asymptotic consistency, and optimality of convergence rate. Aspects of implementation are discussed and the performance characteristics are illustrated by examples.

Computation of the amplitude of stress singular terms for cracks and reentrant corners

Abstract : The theoretical basis and performance characteristics of two new methods for the computation of the coefficients of the terms of asymptotic expansions at reentrant corners from finite element solutions are presented. The methods, called the contour integral method and the cutoff function method, are very efficient: The coefficients converge to their true values as fast as the strain energy, or faster. In order to make the presentation as simple as possible, we assume that the elastic body is homogeneous and isotropic, is loaded by boundary tractions only and, in the neighborhood of the reentrant corner, its boundaries are stress free. The methods described herein can be adapted to cases without such restrictions. Keywords: Finite element methods, P-extension, Fracture mechanics, Elasticity, Stress intensity factors, Mixed mode, Extraction methods, Convergence, Error estimate.

Some recent developments in finite element analysis

Abstract A survey of some recent results in the development of adaptive finite element software systems is presented. The points of view and terminology are those of an engineer-analyst and emphasis is on applications to structural stress analysis.

Mechanical dysfunction in the border zone of an ovine model of left ventricular aneurysm.

BACKGROUND The pathophysiology of regional mechanical dysfunction in the border zone (BZ) region of left ventricular aneurysm was studied in an ovine model using magnetic resonance imaging tissue-tagging and regional deformation analysis. METHODS Transmural infarcts were created in adult Dorsett sheep (n = 8) by ligation of the distal homonymous coronary artery and were allowed to mature into left ventricular aneurysms for 8 to 12 weeks. Animals were imaged subsequently using double oblique magnetic resonance imaging with radiofrequency tissue tagging. Short axis slices were selected for analysis that included predominantly the septal component of the aneurysm as well as adjacent BZ regions in the anterior and posterior ventricular walls. Dark grid patterns of magnetic presaturations were placed on the myocardium and tracked as they deformed during the diastolic, isovolumic systolic, and systolic ejection phases of the cardiac cycle. Regional ventricular wall strains were calculated in BZ regions and regions remote from the aneurysm and compared with strains measured in corresponding regions from normal control sheep (n = 6). RESULTS Diastolic midwall circumferential strains (fiber extensions) were relatively preserved, but abnormal circumferential lengthening strains were observed in the BZ regions during isovolumic systole. Peak circumferential strains ranged from 0.04 to 0.07 in the BZ regions but averaged -0.05 in the normal hearts (p = 0.002 for the anterior BZ and p = 0.001 for the posterior BZ). Midwall end-systolic fiber strains were depressed in the anterior BZ (-0.03 to -0.09 for the BZ versus -0.11 for the normal heart, p < 0.0001) but not in the posterior BZ (p = 0.19). CONCLUSIONS Our data support the theory that the stretching of BZ fibers during isovolumic systole contributed to a reduction in fiber shortening during systolic ejection and thus reduced the overall contribution of these fibers to forward ventricular output.

Quasi-regional mapping for the p-version of the finite element method

In the p-version of the finite element method the size of the elements is fixed independently of the number of degrees of freedom. Therefore, accurate representation of the curves and surfaces which bound the solution domain, so that the quality of the representation is independent of the number of elements, is very important. Another important requirement is that continuous curves and surfaces must be represented either directly, such as in the blending function method, or must be approximated with sufficient accuracy for the discretization error to be controlled by the mesh and the polynomial degree of elements, rather than the mapping of the elements. In this paper we describe a unified representation scheme in which all boundary curves and surfaces are approximated by piecewise polynomials. Special selection of the collocation points provides approximate continuity between elements on smooth boundary curves and surfaces. A numerical example is also presented.

Myocardial material property determination in the in vivo heart using magnetic resonance imaging

Objectives: To determine nonlinear material properties of passive, diastolic myocardium using magnetic resonance imaging (MRI) tissue-tagging, finite element analysis (FEA) and nonlinear optimization.Background: Alterations in the diastolic material properties of myocardium may pre-date the onset of or exist exclusive of systolic ventricular dysfunction in disease states such as hypertrophy and heart failure. Accordingly, significant effort has been expended recently to characterize the material properties of myocardium in diastole. The present study defines a new technique for determining material properties of passive myocardium using finite element (FE) models of the heart, MRI tissue-tagging and nonlinear optimization. This material parameter estimation algorithm is employed to estimate nonlinear material parameters in thein vivo canine heart and provides the necessary framework to study the full complexities of myocardial material behavior in health and disease.Methods and results: Material parameters for a proposed exponential strain energy function were determined by minimizing the least squares difference between FE model-predicted and MRI-measured diastolic strains. Six mongrel dogs underwent MRI imaging with radiofrequency (RF) tissue-tagging. Two-dimensional diastolic strains were measured from the deformations of the MRI tag lines. Finite element models were constructed from early diastolic images and were loaded with the mean early to late left ventricular and right ventricular diastolic change in pressure measured at the time of imaging. A nonlinear optimization algorithm was employed to solve the least squares objective function for the material parameters. Average material parameters for the six dogs wereE=28,722 ± 15,984 dynes/cm2 andc=0.00182 ± 0.00232 cm2/dyne.Conclusion: This parameter estimation algorithm provides the necessary framework for estimating the nonlinear, anisotropic and non-homogeneous material properties of passive myocardium in health and disease in thein vivo beating heart.

Implementation of a C1 triangular element based on the P-version of the finite element method

Abstract The implementation of a computer code CONE (for C1 continuity) based on the p-version of the finite element method is described. A hierarchic family of triangular finite elements of degree p ≥ 5 is used. This family enforces C1-continuity across inter-element boundaries, and the code is applicable to fourth order partial differential equations in two independent variables, in particular to the biharmonic equation. Applications to several benchmark problems in plate bending are presented. Sample results are examined and compared both with theoretical predictions and with the computations of other programs. Significant improvements are shown for the results obtained using CONE.