A Spectral Finite Element Approach to Modeling Soft Solids Excited with High-Frequency Harmonic Loads

An approach for efficient and accurate finite element analysis of harmonically excited soft solids using high-order spectral finite elements is presented and evaluated. The Helmholtz-type equations used to model such systems suffer from additional numerical error known as pollution when excitation frequency becomes high relative to stiffness (i.e. high wave number), which is the case, for example, for soft tissues subject to ultrasound excitations. The use of high-order polynomial elements allows for a reduction in this pollution error, but requires additional consideration to counteract Runge's phenomenon and/or poor linear system conditioning, which has led to the use of spectral element approaches. This work examines in detail the computational benefits and practical applicability of high-order spectral elements for such problems. The spectral elements examined are tensor product elements (i.e. quad or brick elements) of high-order Lagrangian polynomials with non-uniformly distributed Gauss-Lobatto-Legendre nodal points. A shear plane wave example is presented to show the dependence of the accuracy and computational expense of high-order elements on wave number. Then, a convergence study for a viscoelastic acoustic-structure interaction finite element model of an actual ultrasound driven vibroacoustic experiment is shown. The number of degrees of freedom required for a given accuracy level was found to consistently decrease with increasing element order. However, the computationally optimal element order was found to strongly depend on the wave number.     

Numerical inversion of a class of Laplace transforms

Simple formulas are derived to invert a class of Laplace transforms and compute the maximum absolute error. Uniform convergence, simultaneously in the t and s domain is achieved by selecting the approximant's free parameters.     

Frequency sampling method for designing 2-D FIR real-coefficientdigital filters using Hermite interpolation

A novel frequency-sampling method for designing 2-D real-coefficient FIR filters, given the values and slope estimates of the desired frequency response at each of the node points of a rectangular grid, is presented. Based on a new class of bivariate Hermite-type polynomials suitable for interpolating at complex conjugate points, and using Kronecker products, the original 2-D filter design problem is reduced to the solution of two 1-D systems of linear equations. Additional advantages of the method are the securing of the existence and uniqueness of the solution of the design problem, computational efficiency, the use of simple and recursive 1-D algorithms; the guarantee of real accurate results; and the inherent parallelism. The method is also applied to design 2-D symmetric FIR filters and can be extended to m-D design problems     

Least-square-exponential approximation

A method is presented for solving the problem of fitting decay-type experimental data by sums of exponentials. Both the exponents and the coefficients of the exponentials are considered as unknowns. The novel idea presented here is to consider the sum of exponentials as a solution of an integral equation. A step-by-step procedure is given for the solution of the problem in the least-squares sense. An advantage of the proposed method is that the data need not be equidistant.     

A novel method for designing FIR digital filters with nonuniform frequency samples

A new method of designing FIR digital filters using nonuniform frequency samples is presented. There is no restriction on the phase to be linear. The method is based on a Newton-type polynomial interpolating on the unit circle of the complex plane. Attractive features of the proposed method are the applicability to unequally spaced samples, the recursive and semipermanent computation of filter parameters, the capability of obtaining short transition bands or sharp cut-off frequency responses, and the design of efficient algorithms for real-time applications. In the serial case, when the next sample appears, the design parameters are evaluated only by updating the old ones with correction terms that could be used as indicators for convergence, approximation, or filter reduction. The method can be extended to m-D filter design, DFT calculation, design of parallel algorithms, etc.<>