# Sound Field Modeling Method and Key Imaging Technology of an Ultrasonic Phased Array: A Review
## 1 Introduction  
The maturity of array testing theory promotes its application in nondestructive testing. Ultrasonic arrays have two key advantages over standard monolithic sensors. First, a specific array can perform a range of different detections from a single location, making it more flexible than a single-cell sensor. Secondly, the array ultrasound technique allows images to be generated at each test location. This allows rapid visualization of the internal structure of the specimen. These advantages have led to the rapid adoption of arrays in the engineering industry. At present, the research on ultrasonic arrays mainly focuses on transducer development, detection, and imaging methods. A large amount of research has been published in these areas using an array of ultrasound inspection techniques. Modeling plays a crucial role in the use of arrays for Non-Destructive Evaluation (NDE) due to the wide range of array geometries and inspection methods possible. [PERSON] et al. [1] reviewed the modeling approaches which have been adopted to assess the performance of a given array and inspection modality. Both modeling techniques are based on directly emulating the physical operation of an array. However, the development and application of several modeling methods in the non-destructive assessment of ultrasound arrays arenot well reviewed. Based on various detection and imaging modes of ultrasonic arrays, ultrasonic arrays need to study the propagation of sound field in the medium and the response of defects and explore the localization and quantification methods of defects for detection. The modeling methods include analytical approaches, numerical approaches, and semi-analytical approaches. Commonly used analytical calculation approaches include the angular spectrum method (ASM), multi-Gaussian beam method (MGB), and ray tracing method. Numerical approaches include the finite element method (FEM) and finite difference method (FDM). Semi-analytic methods include the distributed point source method (DPSM). Section 2 of this paper gives an overview of the basic theories of the different modeling methods and the application of these methods in ultrasonic arrays by reviewing a wide range of published research. With the development of materials science, the requirements for inspection technology in engineering practice are constantly increasing. The needs for detection of microcracks/micro-defects, detection of complex shaped parts, detection of distal ends of large structures, detection of composite structures, detection of bonding quality, and non-destructive evaluation of the mechanical properties of materials are constantly emerging. Traditional linear ultrasound observation of time domain signals, limited by wavelength and other factors, is not sensitive to micro-defects, micro-cracks, and interface defects such as bonding and delamination. Non-linear ultrasound inspection technology mainly observes the non-linear acoustic response of materials and overcomes the shortcomings of traditional ultrasound by analyzing signal changes in frequency and is an effective complement to traditional linear ultrasound inspection means. [PERSON] et al. [2] reviewed techniques for sizing defects using ultrasonic body waves, including amplitude, time, imaging, and inversion techniques, and the advantages and limitations of various techniques are compared from the perspective of principle and method, but the development and application of nonlinear effects in array ultrasound are not introduced enough. The development and application of several detection techniques in ultrasonic arrays are reviewed in Section 3, including the total focusing method (TFM), compression sensing imaging (CSI), and acoustic nonlinearity imaging (ANI). Section 1 outlines the basic theory, correction algorithm, and acceleration algorithm of TFM as well as reviewing the industry application of the technology.