Songming Hou

A direct imaging algorithm for extended targets

We present a direct imaging algorithm for extended targets. The algorithm is based on a physical factorization of the response matrix of a transducer array. The multi-signal classification imaging function is used to visualize the results. A resolution and noise level-based thresholding strategy is developed for regularization. The algorithm is simple and efficient since no forward solver or iteration is needed. Multiple-frequency information improves both resolution and stability of the algorithm. Efficiency and robustness of the algorithm with respect to measurement noise and random medium fluctuations are demonstrated.

A phase and space coherent direct imaging method

A direct imaging algorithm for point and extended targets is presented. The algorithm is based on a physical factorization of the response matrix of a transducer array. The factorization is used to transform a passive target problem to an active source problem and to extract principal components (tones) in a phase consistent way. The multitone imaging function can superpose multiple tones (spatial diversity/aperture of the array) and frequencies (bandwidth of the probing signal) based on phase coherence. The method is a direct imaging algorithm that is simple and efficient since no forward solver or iteration is needed. Robustness of the algorithm with respect to noise is demonstrated via numerical examples.

Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm

A novel continuation method is presented for solving the inverse medium scattering problem of the Helmholtz equation, which is to reconstruct the shape of the inhomogeneous medium from boundary measurements of the scattered field. The boundary data is assumed to be available at multiple frequencies. Initial guesses are chosen from a direct imaging algorithm, multiple signal classification (MUSIC), along with a level set representation at a certain wavenumber, where the Born approximation may not be valid. Each update via recursive linearization on the wavenumbers is obtained by solving one forward and one adjoint problem of the Helmholtz equation.

Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces

Solving elliptic equations with sharp-edged interfaces is a challenging problem for most existing methods, especially when the solution is highly oscillatory. Nonetheless, it has wide applications in engineering and science. An accurate and efficient method is desired. We propose an efficient non-traditional finite element method with non-body-fitting grids to solve the matrix coefficient elliptic equations with sharp-edged interfaces. Extensive numerical experiments show that this method is second order accurate in the L^~ norm and that it can handle both sharp-edged interface and oscillatory solutions.

A weak formulation for solving elliptic interface problems without body fitted grid

A typical elliptic interface problem is casted as piecewise defined elliptic partial differential equations (PDE) in different regions which are coupled together with interface conditions, such as jumps in solution and flux across the interface. In many situations, such as the interface is moving, the challenge is how to solve such a problem accurately, robustly and efficiently without generating a body fitted mesh. The key issue is how to capture complex geometry of the interface and jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the PDE is not valid across the interface. In this work we present a systematic formulation and further study of a second order accurate numerical method proposed in Hou and Liu (2005) 16] for elliptic interface problem. The key idea is to decompose the solution into two parts, a singular part and a regular part. The singular part captures the interface conditions while the regular part belongs to an appropriate space in the whole domain, which can be solved by a standard finite element formulation. In a general setup the two parts are coupled together. We give an explicit study of the construction of the singular part and the discretized system for the regular part. One key advantage of using weak formulation is that one can avoid assuming/using more regularity than necessary of the solution and the interface. We present the numerical algorithm and numerical tests in 3D to demonstrate the accuracy and other properties of our method.

A direct imaging method using far-field data

We present a direct imaging algorithm for extended targets using far-field data generated by incident plane waves. The algorithm uses a factorization of the response matrix for far-field data that is derived from physical considerations and a resolution-analysis-based regularization. The algorithm is simple and efficient since no forward solver or iteration is needed. Efficiency and robustness of the algorithm with respect to measurement noise are demonstrated.

Recent Studies on Inverse Medium Scattering Problems

Regularized recursive linearization methods are presented for solving a two-dimensional inverse medium scattering problem, which reconstructs the scatterer of an inhomogeneous medium from the measurements of near field current densities. Energy estimates of the scattered field are obtained on which the Born approximation is based. The methods start from the Born approximation corresponding to the weak scattering, each update is obtained via recursive linearization with respect to the wavenumber or the spatial frequency by solving one forward problem and one adjoint problem of the Helmholtz equation. In the case that the weak scattering criterion is not satisfied, a technique based on a direct imaging algorithm may be developed to generate an initial guess. Numerical examples are presented to illustrate the efficiency and robustness of the underlying computational method.

Derivation of the viscous Moore-Greitzer equation for aeroengine flow

The viscous Moore-Greitzer equation modeling the airflow through the compression system in turbomachines, such as a jet engine, is derived using a scaled Navier-Stokes equation. The method utilizes a separation of scale argument based on the different spatial scales in the engine and the different time scales in the flow. The pitch and size of the rotor-stator pair of blades provides a small parameter, which is the size of the local cell. The motion of the stator and rotor blades in the compressor produces a very turbulent flow on a fast time scale. The leading order equation, for the fast time and local scales, describes this turbulent flow. The next order equations produce an axisymmetric swirl and a flow pattern analogous to Rayleigh-Benard convection rolls in Rayleigh-Benard convection. On a much larger spatial scale and a slower time scale, there exist modulations of the flow including instabilities called surge and stall. A higher order equation, in the small parameter, describes these global flow m...

A Numerical Method for Solving 3D Elasticity Equations with Sharp-Edged Interfaces

Interface problems occur frequently when two or more materials meet. Solving elasticity equations with sharp-edged interfaces in three dimensions is a very complicated and challenging problem for most existing methods. There are several difficulties: the coupled elliptic system, the matrix coefficients, the sharp-edged interface, and three dimensions. An accurate and efficient method is desired. In this paper, an efficient nontraditional finite element method with nonbody-fitting grids is proposed to solve elasticity equations with sharp-edged interfaces in three dimensions. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up).

A numerical method for solving three-dimensional elliptic interface problems with triple junction points

Elliptic interface problems with multi-domains have wide applications in engineering and science. However, it is challenging for most existing methods to solve three-dimensional elliptic interface problems with multi-domains due to local geometric complexity, especially for problems with matrix coefficient and sharp-edged interface. There are some recent work in two dimensions for multi-domains and in three dimensions for two domains. However, the extension to three dimensional multi-domain elliptic interface problems is non-trivial. In this paper, we present an efficient non-traditional finite element method with non-body-fitting grids for three-dimensional elliptic interface problems with multi-domains. Numerical experiments show that this method achieves close to second order accurate in the L∞ norm for piecewise smooth solutions.