Assad A. Oberai

Large eddy simulation of turbulent channel flows by the variational multiscale method

The variational multiscale formulation of LES is applied to two-dimensional equilibrium and three-dimensional nonequilibrium channel flows. Simple, constant-coefficient Smagorinsky-type eddy viscosities, without wall damping functions, are used to model the decay of small scales, an approach which is not viable for wall-bounded flows within the traditional LES framework. Nevertheless, very good results are obtained.

The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence

The variational multiscale method is applied to the large eddy simulation (LES) of homogeneous, isotropic flows and compared with the classical Smagorinsky model, the dynamic Smagorinsky model, and direct numerical simulation (DNS) data. Overall, the multiscale method is in better agreement with the DNS data than both the Smagorinsky model and the dynamic Smagorinsky model. The results are somewhat remarkable when one realizes that the multiscale method is almost identical to the Smagorinsky model (the least accurate model!) except for removal of the eddy viscosity from a very small percentage of the lowest modes.

Solution of inverse problems in elasticity imaging using the adjoint method

We consider the problem of determining the shear modulus of a linear-elastic, incompressible medium given boundary data and one component of the displacement field in the entire domain. The problem is derived from applications in quantitative elasticity imaging. We pose the problem as one of minimizing a functional and consider the use of gradient-based algorithms to solve it. In order to calculate the gradient efficiently we develop an algorithm based on the adjoint elasticity operator. The main cost associated with this algorithm is equivalent to solving two forward problems, independent of the number of optimization variables. We present numerical examples that demonstrate the effectiveness of the proposed approach.

Computation of Trailing-Edge Noise Due to Turbulent Flow over an Airfoil

Application of the variational formulation of Lighthill’ s acoustic analogy to trailing-edge noise is considered. Use is made of this formulation to study the effect of e niteness of the chord and the variation of far-e eld pressure directivity with frequency. Numerical analytical solution results are validated for certain limiting cases. Use is also made of this methodology to calculate the far-e eld acoustic pressure for a low-Mach-number turbulent e ow. To determine the acoustic sources for this problem, we employ an unstructured mesh, large eddy simulation of the incompressible Navier ‐Stokes equations.

Evaluation of the adjoint equation based algorithm for elasticity imaging.

Recently a new adjoint equation based iterative method was proposed for evaluating the spatial distribution of the elastic modulus of tissue based on the knowledge of its displacement field under a deformation. In this method the original problem was reformulated as a minimization problem, and a gradient-based optimization algorithm was used to solve it. Significant computational savings were realized by utilizing the solution of the adjoint elasticity equations in calculating the gradient. In this paper, we examine the performance of this method with regard to measures which we believe will impact its eventual clinical use. In particular, we evaluate its abilities to (1) resolve geometrically the complex regions of elevated stiffness; (2) to handle noise levels inherent in typical instrumentation; and (3) to generate three-dimensional elasticity images. For our tests we utilize both synthetic and experimental displacement data, and consider both qualitative and quantitative measures of performance. We conclude that the method is robust and accurate, and a good candidate for clinical application because of its computational speed and efficiency.

Linear and nonlinear elasticity imaging of soft tissue in vivo: demonstration of feasibility.

We establish the feasibility of imaging the linear and nonlinear elastic properties of soft tissue using ultrasound. We report results for breast tissue where it is conjectured that these properties may be used to discern malignant tumors from benign tumors. We consider and compare three different quantities that describe nonlinear behavior, including the variation of strain distribution with overall strain, the variation of the secant modulus with overall applied strain and finally the distribution of the nonlinear parameter in a fully nonlinear hyperelastic model of the breast tissue.

Solution of the nonlinear elasticity imaging inverse problem: the compressible case

We have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material. The data for this problem are the quasi-static deformation fields within the solid measured at two distinct overall strain levels. The main ingredients of our algorithm are a gradient based quasi-Newton minimization strategy, the use of adjoint equations and a novel strategy for continuation in the material parameters. In this paper we present several extensions to this algorithm. First, we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation. We achieve this by solving the forward problem using a residual-based, stabilized, mixed finite element formulation which circumvents the Ladyzenskaya-Babuska-Brezzi condition. Second, we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters, or by splitting the problem into two distinct steps. Finally, we present a new strain energy density function with an exponential stress-strain behavior that yields a deviatoric stress tensor, thereby simplifying the interpretation of pressure when compared with other exponential functions. We test the overall approach by solving for the spatial distribution of material parameters from noisy, synthetic deformation fields.

A multiscale finite element method for the Helmholtz equation

It is well known that when the standard Galerkin method is applied to the Helmholtz equation it exhibits an error in the wavenumber and the solution does not, therefore, preserve the phase characteristics of the exact solution. Improvements on the Galerkin method, including Galerkin least-squares (GLS) methods, have been proposed. However, these approaches rely on a dispersion analysis of the underlying difference stencils in order to reduce error in the solution. In this paper we propose a multiscale finite element for the Helmholtz equation. The method employs a multiscale variational formulation which leads to a subgrid model in which subgrid scales are incorporated analytically through appropriate Greens functions. It is shown that entirely new and accurate methods emerge and that GLS methods can be obtained as special cases of the more general subgrid model.

Computational procedures for determining structural-acoustic response due to hydrodynamic sources

In this paper we propose a methodology for determining the structural-acoustic response due to hydro/aero-dynamic sources. Our methodology is based on solving a fluid problem in which the structure is treated as rigid and using the results of this calculation to solve a structural-acoustic problem. The two problems are coupled using Lighthills acoustic analogy. The key feature of our methodology is the numerical solution of the structural-acoustic problem. For this purpose, we develop new variational formulations of Lighthills acoustic analogy that may be solved using the finite element method. This allows us to consider flexible structures and acoustically non-compact sources in the acoustic calculation with relative ease. We present a preliminary calculation that illustrates the feasibility of our approach.

A residual‐based finite element method for the Helmholtz equation

A new residual-based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter-element boundaries. The inclusion of residuals on inter-element boundaries distinguishes this method from the well-known Galerkin least-squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright