Boundary element analysis of cracks

This paper presents a boundary element formulation that can be used for traction free or pressurized central, edge or branch cracks. The problems associated with the evaluation of integrals containing higher order singularities, which have forced many researchers to modify the boundary integral equations, are overcome in this paper. This permits the use of traction boundary conditions with no modification or addition to the number of unknowns. Comparison of results for conforming and non-conforming elements in modeling of cracks reveals that the oscillation in stresses near the crack tip for non-conforming elements used in the past makes the stress intensity factor dependent on the points used in its calculation. Stress intensity factors are calculated using three different methods. Numerical results for central, edge and branch cracks validate the ideas presented in this paper

Continuity and collocation effects in the boundary element method

The presence of singularities in the integral operators of the boundary element methods requires that the density functions mus 1 satisfy certain continuity requirements if the displacements and stresses are to be bounded. Quite often the continuity conditions, particularly on the derivatives of the density functions, are relaxed at the element ends for the sake of simplicity in approximating the unknown density functions. In this paper, a numerical study on the effects of satisfying or violating the continuity requirements and the effect of the boundary condition collocating point on three different BEM formulations is presented. Two are indirect formulations using force singularities and displacement discontinuity singularities, and the third is Rizzo\u27s direct formulation. The two integral operators in the direct BEM appear individually in the two different formulations of the indirect BEM. This makes it possible to study the numerical error and other problems in each integral operator and the interaction of the two integral operators in the direct BEM. The impact of the study on numerical modelling for the three BEM formulations is presented in the paper

An HR-method of mesh refinement for boundary element method

This paper describes a mesh refinement technique for boundary element method in which the number of elements, the size of elements and the element end location are determined iteratively in order to obtain a user specified accuracy. The method uses L1 norm as a measure of error in the density function and a grading function that ensures that error over each element is the same. The use of grading function along with L1 norm makes the mesh refinement technique applicable to Direct and Indirect boundary element method formulation for a variety of boundary element method applications. Numerical problems in elastostatics, fracture mechanics, and bending of plate solved using Direct and Indirect method in which the density functions are approximated by Linear Lagrange, Quadratic Lagrange or Cubic Hermite polynomials validate the effectiveness of the proposed mesh refinement technique

Optimum interpolation functions for Boundary Element Method

In the Boundary Element Method (BEM) the density functions are approximated by interpolation functions which are chosen to satisfy appropriate continuity requirements. The error of approximation inside an element depends upon the location of the collocation points that are used in constructing the interpolation functions. The location of collocation points also affects the nodal values of the density function and, hence, the total error in the analysis if boundary conditions are satisfied in a collocation sense. In this paper, we minimize the error inside the element using the L1 norm to obtain the optimum location of collocation points. Results show that irrespective of the continuity requirement at the element end, the location of collocation points computed by the algorithm presented in this paper results in an error that is less than the error corresponding to uniformly spaced collocation points. Results for optimum location of collocation points and the average error are presented for Lagrange polynomials up to order fifteen and for Hermite polynomials that ensure continuity up to the seventh order of derivative at the element end. The information of the optimum location of interpolation points for Lagrange and Hermite polynomials should be useful to other researchers in BEM who could incorporate it into their current programs without making significant changes that would be needed for incorporating the algorithm. The algorithm presented is independent of the BEM application in two-dimensions, provided that the density functions are approximated by polynomials and is applicable to direct and indirect formulations. Two numerical examples show the application of the algorithm to an elastostatic problem in which one boundary is represented by integrals of the Direct BEM while the other boundary by the Indirect BEM and a fracture mechanics problem by Direct method in which the crack is represented by displacement discontinuity density function

Do the health benefits of education vary by sociodemographic subgroup? Differential returns to education and implications for health inequities.

PurposeEvidence suggests education is an important life course determinant of health, but few studies examine differential returns to education by sociodemographic subgroup.MethodsUsing National Longitudinal Survey of Youth 1979 (n = 6158) cohort data, we evaluate education attained by age 25 years and physical health (PCS) and mental health component summary scores (MCS) at age 50 years. Race / ethnicity, sex, geography, immigration status, and childhood socioeconomic status (cSES) were evaluated as effect modifiers in birth year adjusted linear regression models.ResultsThe association between education and PCS was large among high cSES respondents (β = 0.81 per year of education, 95% CI: 0.67, 0.94), and larger among low cSES respondents (interaction β = 0.39, 95% CI: 0.06, 0.72). The association between education and MCS was imprecisely estimated among White men (β = 0.44; 95% CI: -0.03, 0.90), while, Black women benefited more from each year of education (interaction β = 0.91; 95% CI: 0.19, 1.64). Similarly, compared to socially advantaged groups, low cSES Blacks, and low and high cSES women benefited more from each year of education, while immigrants benefited less from each year of education.ConclusionsIf causal, increases in educational attainment may reduce some social inequities in health