Gaussian-DPSM (G-DPSM) and Element Source Method (ESM) modifications to DPSM for ultrasonic field modeling Ehsan Kabiri Rahani, Tribikram Kundu ⇑ Department of Civil Engineering and Engineering Mechanics, University of Arizona, P.O. Box 210072, CE Bldg, Tucson, AZ, USA article info Article history: Received 13 December 2010 Received in revised form 11 January 2011 Accepted 11 January 2011 Available online 18 January 2011 Keywords: Distributed Point Source Method Ultrasonic Field Modeling Transducer Modeling Gaussian-DPSM Element Source Method abstract In the last few years, Distributed Point Source Method (DPSM) a mesh-free semi-analytical technique has been developed. In spite of its many advantages, one shortcoming of the conventional DPSM method is that the field obtained by conventional DPSM method needs to be scaled to match the theoretical solu- tions. Two modification techniques called Gaussian-DPSM (G-DPSM) and Element Source Method (ESM) are developed here to avoid the scaling need. G-DPSM technique introduces additional fictitious point sources around every parent point source. Gaussian weight functions determine the strength of these additional fictitious point sources that are denoted as child point sources. ESM replaces discrete point sources used in the conventional DPSM by continuous sources. In the ESM formulation individual point sources are denoted as nodes. Special elements are formed on the boundary by connecting these nodes. The source strength inside the element can vary linearly or non-linearly depending on the order of the interpolation function used inside the element. Results generated by both these methods are compared with the conventional DPSM solution and analytical solution. It is shown that the ultrasonic field in front of the transducer computed by G-DPSM and ESM matches very well with the theory without using any scaling factor. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction In recent years Distributed Point Source Method (DPSM) a semi- analytical technique for modeling ultrasonic and electromagnetic wave propagation problems has gained popularity for modeling Nondestructive Evaluation (NDE) problems. The basic principle of DPSM has been described in detail in Refs. [1–4]. The interaction between ultrasonic waves and fluid–solid interfaces was studied in [5,6]. Ultrasonic field modeling in multilayered fluid structures using DPSM technique was reported by Banerjee et al. [7]. Scatter- ing of ultrasonic waves due to elliptical and spherical cavities has been investigated in Refs. [8,9]. The interaction between bounded ultrasonic beams and corrugated plates has been also studied [10,11]. Electromagnetic acoustic transducers have been modeled using DPSM by Eskandarzade et al. [12]. Kundu et al. [13] compared FEM and DPSM techniques for modeling circular and rectangular transducers. They showed that FEM-3D at high fre- quencies suffers from the lack of accuracy unless the domain is meshed with very small elements that makes this method compu- tationally very time consuming and impossible to handle by many computers while the DPSM technique was much faster and it pro- duced results that matched very well with the analytical solutions. The main shortcoming of DPSM is that when the boundary conditions are satisfied only at the apex points of the spheres (surrounding the point sources) where they touch the transducer surface the computed field needs to be scaled up to match the the- oretical results. This is because the velocity values generated by the point sources are highest at the apex points, therefore the aver- age value of the transducer velocity comes out to be less than the prescribed value if the boundary conditions are satisfied only at the apex points. So far there have been two suggestions to avoid this scaling need – (1) satisfy the boundary conditions at boundary points that are randomly distributed around the apex points, as done in some publications on DPSM [8], and (2) satisfy boundary conditions on and beyond the transducer face at more points than the number of unknowns and thus solve an over-determined system by least-squares error minimization technique as done by Cheng et al. [14]. The problem with option 1 is that random selec- tion of boundary points are not always very reliable and some- times a mismatch between the DPSM results and the theoretical results are observed even after random selection of target points; only after scaling this mismatch can be completely eliminated. The problem with the second option is that it requires solution of an over-determined system and therefore requires more compu- tational effort. In this paper two new modifications are proposed to bypass the scaling needs. In the first modification which will be called Gaussian-DPSM (or G-DPSM) one point source is replaced by 0041-624X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2011.01.004 ⇑ Corresponding author. E-mail address: tkundu@email.arizona.edu (T. Kundu). Ultrasonics 51 (2011) 625–631 Contents lists available at ScienceDirect Ultrasonics journal homepage: www.elsevier.com/locate/ultras