Gauss-Simpson Quadrature Algorithm for Calcaulting Additional Stress in Foundation Soils

The additional pressure at the bottom of a building’s foundation produces an additional stress in the foundation soils under the building’s foundation. In order to overcome the limitations of traditional elastic theory methods and the finite element method when calculating the additional stress in foundation soils, we use the Gauss-Simpson formula to derive the Gauss-Simpson Quadrature Algorithm based on the elasticity theory. The Gauss-Simpson Quadrature Algorithm is a method designed to calculate the additional stress in foundation soils under an irregularly shaped foundation and an irregular load distribution. This new method is based on the fact that the Gaussian quadrature formula and the Simpson formula are independent of the specific type of integrand. The finite element method with n interpolation points can only achieve an algebraic accuracy of n. The interpolation points of the Gaussian quadrature formula are n zeros of orthogonal polynomials, which can achieve an algebraic accuracy of 2n + 1. Moreover, the weights of the nodes in the quadrature formula are all positive, and thus, it has a high numerical stability. In the proposed method, the Simpson formula is necessary. The Simpson formula is used to transform the implicit additional stress formula with the integral sign into an explicit cumulative integral, which can be considered similar to the rectangular domain case to obtain an explicit analytical algebraic formula for solving the additional stress approximation. In engineering applications, we only need to provide the field engineers with the locations of the interpolation points of the Gauss-Legendre formula, the interpolated weight coefficients, and the specific type of Simpson's formula, and then, the results of the additional stress can be calculated manually, which is nearly impossible using the traditional methods and finite element methods. From the point of view of academic rigor and theoretical completeness, it is possible to use the compound Gauss-Simpson Quadrature Algorithm in conjunction with the looping function in computer programs. Under standard conditions, the proposed Gauss-Simpson Quadrature Algorithm is in good agreement with the results of the traditional elasticity theory.

Computation of certain integral formulas involving generalized Wright function

The aim of the paper is to derive certain formulas involving integral transforms and a family of generalized Wright functions, expressed in terms of the generalized Wright hypergeometric function and in terms of the generalized hypergeometric function as well. Based on the main results, some integral formulas involving different special functions connected with the generalized Wright function are explicitly established as special cases for different values of the parameters. Moreover, a Gaussian quadrature formula has been used to compute the integrals and compare with the main results by using graphical representations.

Efficient Random Field Uncertainty Propagation in Design Using Multiscale Analysis

An integrated design framework that employs multiscale analysis to facilitate concurrent product, material, and manufacturing process design is presented in this work. To account for uncertainties associated with material structures and their impact on product performance across multiple scales, efficient computational techniques are developed for propagating material uncertainty with random field representation. Random field is employed to realistically model the uncertainty existing in material microstructure, which spatially varies in a product inherited from the manufacturing process. To reduce the dimensionality of random field representation, a reduced order Karhunen–Loeve expansion is used with a discretization scheme applied to finite-element meshes. The univariate dimension reduction method and the Gaussian quadrature formula are used to efficiently quantify the uncertainties in product performance in terms of its statistical moments, which are critical information for design under uncertainty. A control arm example is used to demonstrate the proposed approach. The impact of the initial microscale porosity random field produced during a casting process on the product damage is studied and a reliability-based design of the control arm is performed.

A Multiscale Design Approach With Random Field Representation of Material Uncertainty

A multiscale design approach is proposed in this paper considering the impacts of product manufacturing process and material on product performance. A framework is established to integrate designs of manufacturing process, material and product based on the information flow across these three domains. Random field is employed to realistically model the uncertainty existing in material microstructure which spatially varies in a product inherited from the manufacturing process. An efficient procedure for uncertainty propagation from the material random field to the end product performance is established. To reduce the dimensionality of random field representation, a reduced order Karhunen-Loeve expansion is used with a discretization scheme applied to finite element meshes. The univariate dimension reduction method and the Gaussian quadrature formula are used to efficiently quantify the uncertainties in product performance in terms of its statistical moments, which are critical information for design under uncertainty. A control arm example is used to demonstrate the proposed approach. The impact of the initial microscale porosity random field produced during a casting process on the product damage is studied and a reliability-based design of the control arm is performed.

On the Drift Forces of a Vertical Cylinder in Water of Finite Depth

Analytical solutions for the wave-induced second-order time independent drift forces and moments due to the dynamic and the waterline pressures on a fixed vertical circular cylinder are derived. The results are displayed graphically for a number of depth to radius ratios. An analytical technique is used to determine the first-order velocity potential by considering two regions, namely, interior region and exterior region. We have also demonstrated a numerical solution by a higher order panel method in which the kernel of the integral equation is modified to make it non-singular and amenable to solutions by the Gaussian quadrature formula. The numerical results are found to comply with the analytical solutions.