T. A. Cruse

Numerical solutions in three dimensional elastostatics

Abstract A numerical solution capability is developed for the solution of problems in three dimensional elastostatics. The solution method utilizes singular integral equations which can be solved numerically for the unknown surface tractions and displacements for the fully mixed boundary value problem. The method is independent of the surface shape and data specification and has been fully automated. Some sample problems are solved to verify the formulation. In addition the method has been used to investigate a significant problem with stress singularities.

Numerical solutions in axisymmetric elasticity

Abstract Paper presents the formulation of the axisymmetric elasticity problem with thermal and rotational loading using the boundary-integral equation method. The resulting one dimensional numerical model is evaluated for a series of problems, including a problem solved by a finite element model. Results show the validity of the formulation for non-trivial problems, as well as the advantage of significant modeling efficiency relative to the finite element method for certain classes of problems.

Traction BIE solutions for flat cracks

The paper deals with the numerical solution techniques for the traction boundary integral equation (BIE), which describes the opening (and sliding) displacements of the surface of the traction loaded crack or arbitrary planform embedded in an elastic infinite body (buried crack problem). The traction BIE is a singular integral equation of the first kind for the displacement gradients. Its solution poses a number of numerical problems, such as the presence of derivatives of the unknown function in the integral equation, the modeling of the crack front displacement gradient singularity, and the regularization of the equations singular kernels. All of the above problems have been addressed and solved. Details of the algorithm are provided. Numerical results of a number of crack configurations are presented, demonstrating high accuracy of the method.

Probabilistic Structural Analysis Methods for select space propulsion system structural components (PSAM)

Abstract The purpose of doing probabilistic structural analysis is to provide the designer with a more realistic ability to assess the importance of uncertainty in the response of a high performance structure. The objective of this five-year contract effort (NASA Contrast NAS3-24389) on Probabilistic Structural Analysis Methods (PSAM) is the development of several modular structural analysis packages capable of predicting the probabilistic response distribution for key structural variables such as stress, displacement, natural frequencies, buckling loads, transient responses, etc. The structural analysis packages include stochastic modeling of loads, material properties, geometry (e.g. tolerances), and boundary conditions. The structural analysis solution is in terms of the cumulative distribution function (CDF) and confidence bounds. Two methods of probability modeling are included in the analysis packages. One of these is the well established Monte Carlo simulation method; the other is a fast probability integration (FPI) algorithm which will be discussed in the paper. PSAM tools can be used to estimate structural safety and reliability, while providing the engineer with information on the confidence that should be given to the predicted behavior. Perhaps most critically, the PSAM results provide information to the designer on those variables for which the design is most sensitive.

On the Somigliana stress identity in elasticity

The paper presents and analytical investigation of the properties of the integral terms in the Somigliana stress identity which are associated with a perceived hypersingular nature of the integral equations in three dimensional elasticity. The nature of the integral equations is, in fact, found to be non-hypersingular, thereby permitting direct evaluation of the jump terms in the stress identity for a solution point taken, in the limit, to the surface of the body. The continuity requirements on the boundary conditions are found to be more liberal than previously reported. A weakly-singular form of the Somigliana identity is found that is easily used for BEM implementations that use Gaussian integrations. Demonstration of the boundary form of the Somigliana stress identity is given for a three dimensional elasticity problem.

PROBABILISTIC STRUCTURAL ANALYSIS METHODS AND APPLICATIONS

Abstract An advanced algorithm for simulating the probabilistic distribution of structural responses due to statistical uncertainties in loads, geometry, material properties and boundary conditions is reported. The method effectively combines an advanced algorithm for calculating probability levels for multivariate problems—Fast Probability Integration (FPI)—together with a general purpose finite element code for stress, vibration and buckling analysis. The combination provides the analyst with the ability to calculate probability levels for output variables with speed and accuracy over the full range of probability levels. Application is made to a space propulsion system turbine blade for which the geometry and material properties are treated as random variables. The results also demonstrate the ability of the code to define the relative ranking of the random variables in terms of their influence on the response variable uncertainty. Extension of the code to nonlinear problems and to other analysis modules is being made.

WEAKLY SINGULAR STRESS-BEM FOR 2D ELASTOSTATICS

A weakly singular stress-BEM is presented in which the linear state regularizing field is extended over the entire surface. The algorithm employs standard conforming C0 elements with Lagrangian interpolations and exclusively uses Gaussian integration without any transformation of the integrands other than the usual mapping into the intrinsic space. The linear state stress-BIE on which the algorithm is based has no free term so that the BEM treatment of external corners requires no special consideration other than to admit traction discontinuities. The self-regularizing nature of the Somigliana stress identity is demonstrated to produce a very simple and effective method for computing stresses which gives excellent numerical results for all points in the body including boundary points and interior points which may be arbitrarily close to a boundary. A key observation is the relation between BIE density functions and successful interpolation orders. Numerical results for two dimensions show that the use of quartic interpolations is required for algorithms employing regularization over an entire surface to show comparable accuracy to algorithms using local regularization and quadratic interpolations. Additionally, the numerical results show that there is no general correlation between discontinuities in elemental displacement gradients and solution accuracy either in terms of unknown boundary data or interior solutions near element junctions. Copyright

The Boundary Element Method

Diffusion problems - corrosion cells and first estimates of crevice corrosion behaviour acoustics - the Boundary Element Method applied to simulation of active vibration control heat transfer electromagnetics - a BEM spproach for grounding grid computation inverse problems - solving inverse heat transfer problems by BEM numerical and computational aspects - the analog equation method, a powerful BEM-based solution technique for solving linear and non-linear engineering problems adaptive techniques, accuracy and error analysis stress analysis - boundary element analysis of Cosserat Continuum contact mechanics - damage tolerance, modelling requirements for fracture mechanics vibrations - substructure deletion method, a boundary element approach for elastodynamic problems geomechanics - modelling of non-linear soil structure interface behaviour using BEM coupling problems - a combined boundary element and finite strip solution, BSM.

Nonsingular BEM for fracture modeling

Abstract A new three-dimensional formulation for the fracture mechanics problem based on a non-singular form of the hypersingular Somigliana stress identity (SSI) is presented. The resulting formulation permits the use of low-order numerical integration algorithms and allows the crack opening displacement variable to be modeled in a piecewise- C 1, α manner which is well suited to standard boundary element method (BEM) algorithms. The method is not limited to particular crack shapes and is sufficiently general that it holds excellent prospects for providing a general class of numerical Greens functions for this important class of problems.

A general solution procedure for fracture mechanics weight function evalution based on the boundary element method

This paper reports on the development of an efficient and accurate means for the direct computation of crack surface weight functions for two dimensional fracture mechanics analysis. Weight functions are mathematical representations which can be used to efficiently calculate stress intensity factors for a variety of crack loading and boundary conditions. The method is inherently capable of handling mixed-mode problems. The weight function capability is especially important for problems of fatigue crack growth modeling where the efficient calculation of stress intensity factors is crucial.The basis of the new formulation and numerical solution method is the boundary element method (BEM), as implemented for two dimensional fracture mechanics analysis. The paper will review the analytical formulation of the new BEM, the numerical solution algorithm, and a limited number of validation examples.