# Model of wave propagation in anisotropic media of dissimilar and multi-layer welded joints

## Description

The purpose of this study is to verify the numerical simulation of the finite difference method of the time of flight diffraction (TOFD) in studies of welded joints and multi-layer models, for the case of multi-layer welded joints. In this way, the effects of a type of anisotropy, specifically transverse isotropy (system with elastic hexagonal or polar symmetry) can be evaluated in the propagation of waves in elastic media by ultrasonic inspection. The algorithm for the construction of the model applies the finite difference method and was created in FORTRAN language. The evaluated approach is in two-dimensional space. The evaluated case is composed of a cross-sectional model of a welded joint similar to the one shown in the Figure below, with the insertion of discontinuities to the material. This case aims mainly to evaluate the propagation behavior of the wave in relation to its interaction with discontinuities of different sizes and positions within the material. Thus, through the response of A-Scan signals (echogram), it was possible to evaluate the relevant parameters to obtain a satisfactory non-destructive inspection. The models of the welded joints are formed by a plane of depth xz with 5000 x 1500 points in the respective directions. For practical purposes, spatial sampling rules according to domain size were approximated in terms of the total number of arithmetic operations in finite difference schemes of higher order; for this, five points were used for the smallest wavelength defined by the minimum velocity in the model. In this way, all the simulated images (components of the displacement field and cross sections) are presented with a factor of five of mesh contraction, that is, the sample grid with 5000 x 1500 points becomes 1000 x 300 points (indicated in the figures). ). This transformation is important for the reduction of the arithmetic operations involved and in the computational time spent for the compilation of the elastic property files, but it does not present any visual alteration in the presentation of these images. The temporal sampling interval adopted for this case was 35.0s with temporal resolution of 4.7ns. In the simulations with the model (welded joints), A-Scan echograms were generated. The results are attached for wide dissemination.

## Files

## Steps to reproduce

A vertical crack with defined dimensions was added to the weld joint model, as well as the schematic presentation of the Figure below. This resulted in 64 different situations of parameter variations, named according to the rules of the Table below, where a combination of: four values for depth c (5.0 / 10.0 / 15.0 / 20.0 [mm]), four values for length b (2.0 / 4.0 / 6.0 / 8.0 [mm]) and four values for thickness a (0.5 / 1.0 / 1.5 / 2.0 [mm]), according to a fixed thickness d of 30.0mm. The geometric configuration of the welded joint and the internal defects is arranged inside the mesh according to the properties of each region. To each region of the welded joint grid is assigned a group of elastic properties for each material of its composition, according to the characteristics raised (see related links). With all properties entered in the model, the propagation process is ready to start. The simulation of wave propagation begins with an impulsive energy source represented by an analytical function. This source was inserted on the surface of the model, as well as the receiving transducer. Both are arranged in a standard TOFD configuration. The spacing between the transducers is 80mm. The source used for the numerical simulation is a directional type with 45 ° angle and 5MHz frequency, located on the surface of the model in a standard TOFD array with receiver also on the surface, but on the opposite side. The process of wave propagation occurs initially through a perturbation in initial condition through an analytical excitation defined by the second-order derivative of the Gaussian function. With the explosion of the source, propagation begins. As the finite difference method is based on the elastic wave equation and not on particular solutions, it includes both direct waves (P and S) as well as surface waves, refracted, diffracted and converted; and still preserves the amplitude relations of events. The data in attachments exposes the simulated results by: (i) the images of the displacement field components in 14.0s, (ii) the cross-sections (macrography) simulated with defect size and (iii) the echogram of A-Scan type. With the images of the displacement field components, it is possible not only to obtain the format of the propagated P and S wave fronts but also the numerical evaluation of the signal amplitude response through the A-Scan analysis with transducers, emitter and receiver , arranged on the surface of the model in a standard TOFD configuration.