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Engineering systems are often too complex and their behavior cannot be determined using closed form or exact methods. To circumvent this problem, numerical methods, such as boundary element method, have been formulated to obtain approximate solutions to partial differential equations, which describe the behavior of the physical engineering systems. However, the inherent uncertainty in constitutive formulation causes uncertainties in the solutions obtained by numerical methods and undermines their validity. Conventional analysis does not account for material uncertainty, which is, however, accounted for in the design phase. This paper addresses the impact of uncertain shear modulus for plane strain linear elasticity problems on the numerical solutions obtained using boundary element method. The uncertainty is modeled using fuzzy approach. Matrix parameterization is developed to obtain exact worst case bounds on the solutions assuming that a correct partial membership function is given.

The response of the engineering system is often obtained by the use of numerical methods such as finite element method or boundary element method. However, the uncertainty of the acquired solutions cannot be measured using conventional methods. This uncertainty is attributed to two sources: errors in mathematical modeling and uncertainties in the parameter. The following paper addresses the second source of uncertainty for the steady state heat conduction problem where the material conductivity is uncertain. Material uncertainty is implemented into fuzzy boundary element method which obtains the exact worst case bounds on the response given the worst case bounds on the parameter uncertainty. The method assumes that a correct partial membership function is given. Numerical examples are shown to illustrate the behavior of the method.

Solutions to partial differential equations describing behavior of physical systems are often imprecise. This uncertainty is due to numerical approximations and uncertainty in physical parameters. In elastostatics, these parameters include uncertain material behavior, uncertain boundary conditions, and uncertain geometry of the system. This paper addresses the treatment of geometrical uncertainty for elasticity problems. The new method predicts fuzzy responses for the given membership functions, describing the range of tolerances for the system’s geometry. To obtain exact bounds on the solution to the resulting fuzzy linear system of equations, fuzzy matrix parameterization is developed. Numerical example is shown to illustrate the behavior of the method.

In engineering most partial differential equations are solved using numerical methods. Throughout the years finite element method emerged as the most widely used numerical technique to obtain discrete solutions to partial differential equations. An alternative method to the finite element method is the boundary element method. In boundary element formulation the domain variables are transformed to boundary variables using Green's functions of the partial differential equations. The uncertainty in the solutions to boundary values in boundary element method has been studied using interval approach. Interval treatment of the uncertainty in boundary conditions, integration, truncation, and discretization errors has been developed. In this work local discretization error is computed for the problems exhibiting singular flux. Example is shown demonstrating the behavior of the worst case bounds on the discretization error in the presence of singular solutions.

Reliable design is dependent on reliable engineering simulation. Many problems of engineering analysis rely on numerical techniques for solving partial differential equations. The foremost method of obtaining approximate solution to partial differential equations is the finite element method. However, alternate approaches, such as the boundary element method, have been proven to be more accurate in certain types of problems. In this work, a new method is developed to quantify the discretization error in boundary element analysis. The method considers the discretization error as an interval variable and a scheme is developed to obtain sharp results of the interval guaranteed to bound the solution. Thus, the error bounds are in terms of worst case bounds that can be used directly in design decisions. Exemplars are presented showing that the obtained interval solutions enclose the closed form solutions.