We study the application of a B-splines Finite Element Method (FEM) to time-harmonic scattering acoustic problems. The infinite space is truncated by a fictitious boundary and second-order Absorbing Boundary Conditions (ABCs) are applied. The truncation error is included in the exact solution so that the reported error is an indicator of the performance of the numerical method, in particular of the size of the pollution error. Numerical results performed with high-order basis functions (third or fourth order) showed no visible pollution error even for very high frequencies. To prove the ability of the method to increase its accuracy in the high frequency regime, we show how to implement a high-order Padé-type ABC on the fictitious outer boundary. The above-mentioned properties combined with exact geometrical representation make B-Spline FEM a very promising platform to solve high-frequency acoustic problems.
This work is concerned with a unique combination of high order local absorbing boundary conditions (ABC) with a general curvilinear Finite Element Method (FEM) and its implementation in Isogeometric Analysis (IGA) for time-harmonic acoustic waves. The ABC employed were recently devised by Villamizar, Acosta and Dastrup [J. Comput. Phys. 333 (2017) 331] . They are derived from exact Farfield Expansions representations of the outgoing waves in the exterior of the regions enclosed by the artificial boundary. As a consequence, the error due to the ABC on the artificial boundary can be reduced conveniently such that the dominant error comes from the volume discretization method used in the interior of the computational domain. Reciprocally, the error in the interior can be made as small as the error at the artificial boundary by appropriate implementation of p-and h-refinement. We apply this novel method to cylindrical, spherical and arbitrary shape scatterers including a prototype submarine. Our numerical results exhibits spectral-like approximation and high order convergence rate. Additionally, they show that the proposed method can reduce both the pollution and artificial boundary errors to negligible levels even in very lowand high-frequency regimes with rather coarse discretization densities in the IGA. As a result, we have developed a highly accurate computational platform to numerically solve time-harmonic acoustic wave scattering in two-and three-dimensions.
The human foot is subjected to ground reaction forces during running. These forces have been studied for decades to reduce the related injuries and increase comfort. A four-degree-of-freedom system has been used in the literature to simulate the human body motion during the touch-down. However, there are still inconsistencies between the simulation results and experimental measurements. In this study, an optimization technique is proposed to obtain the required parameters to estimate the vertical ground reaction force using the measurements from actual runners. The touch-down velocities of the rigid and wobbling body masses were also treated as optimization variables. It was shown that the proposed parameters can be adjusted to represent a particular shoe type. Specifically, vertical ground reaction force parameters and touch-down velocities were obtained for shoes with various insole properties and cushioning technologies. The results of this study suggest that the human locomotion system reacts to the shoe properties by regulating the velocities of the body wobbling and rigid masses. The magnitude and the load rate obtained using the proposed parameters are consistent with the experimental data. It is shown that the viscoelastic properties of the shoe will significantly affect the load rate but not the load magnitude.
The formulation of high order local absorbing boundary conditions in terms of farfield expansions (FE-ABC) for time-harmonic waves is considered. Our previously constructed FE-ABC for single and multiple acoustic scattering are improved. Furthermore, we extend the FE-ABC to elastic scattering. By decomposing the vector elastic displacement in terms of scalar potentials, the Navier's equation of elasticity is reduced to a boundary value problem consisting of two Helmholtz equations coupled through their boundary conditions. As a result, the formulation of the elastic FE-ABC from the acoustic case becomes natural. In this work, we present some numerical results by coupling the FE-ABC with finite difference methods and a general curvilinear finite element method based on isogeometric analysis. We present our results for two and three dimensional acoustic and elastic scattering problems from obstacles of arbitrary shape.
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