Thomas J. Rudolphi

Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations

Some integral identities for the fundamental solutions of potential and elastostatic problems are established in this paper. With these identities it is shown that the conventional boundary integral equation (BIE), which is generally expressed in terms of singular integrals in the sense of the Cauchy principal value (CPV), and the derivative BIE, which is similarly expressed in terms of hypersingular integrals in the sense of the Hadamard finite-part (HFP), can both be written as weakly-singular integral equations in a systematic approach. Discretization of the weakly-singular BIE leads to the weakly-singular boundary element formulation equivalent to the method of using the rigid body displacement to determine the diagonal submatrices, which involve the CPV terms and the geometric matrix C, in the conventional BEM. The discretization of the weakly-singular derivative BIE possesses a similar feature, i.e. no CPV and HFP are involved. All these suggest that the practice of calculating CPV or HFP (for boundary integrals) and the geometric matrix C, either analytically or numerically, is unnecessary in the BEM. The approach developed in this paper is applicable to other problems such as plate bending, acoustics and elastodynamics.

Effects of stride length and running mileage on a probabilistic stress fracture model.

UNLABELLED The fatigue life of bone is inversely related to strain magnitude. Decreasing stride length is a potential mechanism of strain reduction during running. If stride length is decreased, the number of loading cycles will increase for a given mileage. It is unclear if increased loading cycles are detrimental to skeletal health despite reductions in strain. PURPOSE To determine the effects of stride length and running mileage on the probability of tibial stress fracture. METHODS Ten male subjects ran overground at their preferred running velocity during two conditions: preferred stride length and 10% reduction in preferred stride length. Force platform and kinematic data were collected concurrently. A combination of experimental and musculoskeletal modeling techniques was used to determine joint contact forces acting on the distal tibia. Peak instantaneous joint contact forces served as inputs to a finite element model to estimate tibial strains during stance. Stress fracture probability for stride length conditions and three running mileages (3, 5, and 7 miles x d(-1)) were determined using a probabilistic model of bone damage, repair, and adaptation. Differences in stress fracture probability were compared between conditions using a 2 x 3 repeated-measures ANOVA. RESULTS The main effects of stride length (P = 0.017) and running mileage (P = 0.001) were significant. Reducing stride length decreased the probability of stress fracture by 3% to 6%. Increasing running mileage increased the probability of stress fracture by 4% to 10%. CONCLUSIONS Results suggest that strain magnitude plays a more important role in stress fracture development than the total number of loading cycles. Runners wishing to decrease their probability for tibial stress fracture may benefit from a 10% reduction in stride length.

Continuity requirements for density functions in the boundary integral equation method

Two methods of forming regular or hypersingular boundary integral equations starting from an interior integral representations are discussed. One method involves direct treatment of the singularities such as Cauchy principal value and/or finite-part interpretation of the integrals and the other does not. By either approach, theory places the same restrictions on the smoothness of the density function for the integrals to exist, assuming sufficient smoothness of the geometrical boundary itself. Specifically, necessary conditions on the smoothness of the density function for meaningful boundary integral formulas to exist as required for the collocation procedure are established here. Cases for which such conditions may not be sufficient are also mentioned and it is understood that with Galerkin techniques, weaker smoothness requirements may pertain. Finally, the bearing of these issues on the choice of boundary elements, to numerically solve a hypersingular boundary integral equation, is explored and numerical examples in 2D are presented.

Effects of running speed on a probabilistic stress fracture model.

BACKGROUND Stress fractures are dependent on both loading magnitude and loading exposure. Decreasing speed is a potential mechanism of strain reduction during running. However, if running speed is decreased the number of loading cycles will increase for a given mileage. It is unclear if these increased loading cycles are detrimental despite reductions in bone strain. The purpose of this study was to determine the effects of running speed on the probability of tibial stress fracture during a new running regimen. METHODS Ten male subjects ran overground at 2.5, 3.5, and 4.5m/s. Force platform and kinematic data were collected synchronously. Inverse dynamics and musculoskeletal modeling were used to determine joint contact forces acting on the distal tibia. Peak tibial contact force served as input to a finite element model to estimate tibial strains. Stress fracture probability for each running speed was determined using a probabilistic model based on published relationships of bone damage, repair, and adaptation. The effects of speed on stress fracture probability was compared using a repeated measures ANOVA. FINDINGS Decreasing running speed from 4.5 to 3.5m/s reduced the estimated likelihood for stress fracture by 7% (P=0.017). Decreasing running speed from 3.5 to 2.5m/s further reduced the likelihood for stress fracture by 10% (P<0.001). INTERPRETATION Runners wanting to reduce their risk for tibial stress fracture may benefit from a decrease in running speed. For the speeds and mileage relative to the current study, stress fracture development was more dependent on loading magnitude rather than loading exposure.

Hypersingular Boundary Integral Equations: A New Approach to Their Numerical Treatment

In the first part, hypersingular boundary integral equations are obtained through proper consideration of the limiting process. It is proved that no divergent terms actually arise, and that interpretations of the integrals are not required. In the second part, a general algorithm for the direct numerical treatment of hypersingular integrals in the BEM is developed. The proposed approach operates in terms of intrinsic coordinates and shows any hypersingular integral in the BEM to be equivalent to a sum of two regular integrals. Numerical results on curved elements are presented.

Viscoelastic Indentation and Resistance to Motion of Conveyor Belts using a Generalized Maxwell Model of the Backing Material

A one-dimensional Winkler foundation and a generalized viscoelastic Maxwell solid model of the belt backing material are used determine the resistance to motion of a conveyor belt over idlers. The viscoelastic material model is a generalization of the three-parameter Maxwell model that has previously been used to predict the effective frictional coefficient of the rolling motion. Frequency, or loading rate, and temperature dependence of the material properties are incorporated with the time/temperature correspondence principle of linear viscoelastic materials. As a consequence of the Winkler foundation model, a normalized indentation resistance is independent of the primary belt system parameters – carrying weight per unit width, idler diameter and backing thickness - as is the case for a three-parameter viscoelastic model. Example results are provided for a typical rubber compound backing material and belt system parameters.

A boundary element model for acoustic-elastic interaction with applications in ultrasonic NDE

The boundary integral equation method is applied to a class of time-harmonic acoustic scattering problems where the bounded elastic scatterer is submerged in a fluid. An exact mathematical model is presented for a finite scatterer with a closed surface, where the surface integral equations are exclusively used to represent the fluid-solid or acoustic-elastic interaction of the scattering process. The numerical procedure involves application of point collocation with quadratic isoparametric approximations that reduce the integral equations to a discrete set of complex linear algebraic equations. Examples emphasize the potential of the method to solve three-dimensional problems of practical interest. Limitations of the formulations and the extension to the case of a semi-infinite plane and curved fluid-solid interface are discussed in the latter part of the paper.

A weakly singular formulation of traction and tangent derivative boundary integral equations in three dimensional elasticity

Abstract Regularized forms of the traction and tangent derivative boundary integral equations are derived for three dimensional elasticity. Kernels of the resulting equations contain only weak singularities and thus are amenable to the ordinary numerical treatment required by weakly singular integrals. The hypersingular and strongly singular kernels of the displacement gradient representation are regularized independently, through identities of the fundamental solution and its various derivatives, before the integral equations are formed. The stress or traction equations provide an alternate formulation of boundary value problems in the same boundary variables as the displacement boundary integral equations, and are useful for problems where the displacement equations are deficient, such as in crack problems. Alternatively, the tangent derivative equations explicitly introduce surface displacement derivatives and are thus completely independent equations that may be used simultaneously with the displacement or traction equations.

Discretization Considerations with Hypersingular Integral Formulas for Crack Problems

A hypersingular integral equation is derived to solve the cracklike problem of acoustic scattering from a thin rigid screen. Discretization considerations regarding modeling the problem and computing the integrals to a required precision, using the Boundary Element Method (BEM), are discussed. Numerical results are presented.

Solution of potential problems using combinations of the regular and derivative boundary integral equations

Abstract The feasibility of using combinations of the boundary integral equation (BIE) and the normal derivative of the boundary integral equation (DBIE) is investigated for the two-dimensional Laplace equation. By using the combinations of these two equations it is possible to derive Fredholm integral equations of either the first or second kind regardless of the boundary conditions. Although the Fredholm equations of the second kind are well conditioned, in general they provide less accurate interior results than the Fredholm equations of the first kind and the classical direct boundary element method (DBEM). On the other hand, the Fredholm equations of the first kind can provide poor solutions in regions close to the boundary where large gradients exist in the boundary flux.