Reconstructing two-dimensional acoustic object fields by use of digital phase conjugation of scanning laser vibrometry recordings

A scanning laser Doppler vibrometer is used to record two-dimensional ultrasound fields in air. The laser light of the vibrometer traverses the sound field to and from a rigid reflector and determines the velocity field, a quantity proportional to the sound pressure rate, in each scanned point relative to the sound source. The object sound is the scattered field from objects outside the recording area. Digital reconstruction by use of phase conjugation (time reversal) of the object sound field is then performed, and the original object field intensity and phase is reconstructed.     

Formulation and solution of hierarchical finite element equations

The paper presents a general hierarchical formulation applicable to both elliptic and hyperbolic problems. Static and eigenvalue linear elastic problems as well as convection–diffusion problems are studied. The hierarchical formulation is well suited for adaptive procedures. For the convection-diffusion problem the hierarchical approximation is made in time only. Different hierarchical functions are proposed for different types of problems. Both weighted residual and least-squares formulations are applied. A combination of these two gives a penalty method with a constraint equation corresponding to the least-squares method. A whole class of time integration formulae is obtained. These are all suitable for adaptive procedures owing to the hierarchical approximation in the time domain. If a linear discontinuous hierarchical base function is used in the Galerkin weak formulation, the method so obtained corresponds to the discontinuous Galerkin method in time and is especially suited for convection dominated problems. The streamline-diffusion method is found to be the aforementioned penalty method. This paper also examines the sequence of nested equation systems that results from a hierarchical finite element formulation. Properties of these systems arising from static problems are investigated. The paper presents some new possibilities for iterative solution of hierarchic element equations, and different procedures are compared in a numerical example. Finally, a simple ID convection-diffusion problem clearly shows that the proposed hierarchical formulation in time gives a stable and accurate solution even for convection dominated flow.