Finite Element theory

FEA is commonly used to investigate the behaviour of complex systems that change over time. The complexity in these systems could be in the geometry, the wave interaction behaviour such as the level of scattering, reflection or mode conversion, or in the changes that such a system sees with time such that an analytical solution to the problem is unavailable. For example in the problem that is presented here, the phase and group speeds of the different modes are given by the respective dispersion curves as in Chapter 2, but the extent to which waves are reflected or mode converted can’t be derived from these dispersion curves.

The finite element method is based on building a structure out of a series of discrete elements such that a large sample is constructed from many small elements, this is known as building a mesh [145], and an example of this is shown in Figure 4.1. This mesh has to be fine enough to allow for a good approximation of the overall

structure to be achieved. The small elements that the sample is split into allows for the wave equation for the situation to be solved for each individual element and the results propagated into the next element, whilst also ensuring global boundary conditions are met. In modelling processes for NDT tasks, a chronological solution to the problem is found usually employing a time stepping solution. Breaking the problem down in this way allows for very complex systems to be examined due to the calculations for each element being simple and easy to calculate but the combination of all these solutions making up the more complex global solution.

Numerically solving the wave equations for the mesh of elements provides a self consistent series of solutions for each element, which is used to obtain a global solution to the wave equation for the whole model of which the elements are a part, and this is then used to calculate the wave displacements, velocities or pressures, as is required for a given situation.

Figure 4.1: Illustration of the concept of meshing in finite element methods, with many small elements making up a larger model. The solutions for the wave equation are found for each element allowing the propagation of waves through the model.

When designing a finite element simulation, care must be taken that the small errors that can be inherent in building a large model out of smaller elements do not add up to make the output of the model non representative of the experimental situation. These errors are generally introduced by rounding errors or mistakes that are made when discretising the model. If the mesh is not fine enough then a propagating wave will experience numerical dispersion [146, 147]. The mesh density that is required for a given situation in order to accurately represent the sample is generally found through convergence of the solution. Although various suggestions for the number of elements that are required can be seen in the literature, in reality a convergence test is the only method that will provide a definitive answer [145, 148, 149].

The convergence of a model is the concept that as the size of the elements is reduced, hence increasing the density of the mesh for a given distance, the solution to the finite element problem becomes closer to the actual experimental results. For

example in an ultrasonic case, the speed of the ultrasonic waveforms in the simu- lation should approach the analytical value for that type of wave. A common way of confirming the convergence of the solution is by doubling the mesh density and investigating whether the output of the model changes significantly. Here conver- gence was tested by testing the arrival time of the modes to check whether they were close to the theoretical values. If the theoretical arrival time value wasn’t reached, then the number of elements was doubled and the model rerun. This procedure continued until the solution converged. This is illustrated in Figure 4.2 where the number of elements that make up the distance represented by one wavelength of the ultrasound generated is doubled from one case to the next, with the extra elements in that distance providing a more representative solution to the problem.

Figure 4.2: Illustrative example of doubling the number of elements in a model for the distance given by one wavelength of the ultrasound in the model. Convergence is achieved when the signal representation in the model doesn’t improve when the number of elements is increased.

The convergence test is easy to carry out on a flat plate style geometry due to the simplicity of the propagation of the waves between the elements but becomes more complex when a pipe structure is considered. This is because PZFlex as a finite element analysis tool doesn’t provide the user with a choice of element geometries, it limits the mesh choice to that of square or cubic elements. This can cause issues with generating geometries that are more complex such as curved surfaces as the number of elements will have to be increased to bring the approximation of the surface closer to the real situation, the issue of a square mesh for a circular sample is illustrated in Figure 4.3. This limiting factor is avoided in other finite element programs where a choice of the mesh type is allowed so that a suitable element shape

can be chosen to suit the geometry and a variation in mesh density is possible such that areas of interest can be finely meshed and areas that are not as relevant to the solution can be more broadly meshed. This allows for the mesh to provide a closer approximation to complex geometries such as curved surfaces by using elements that can be distorted to match the desired surface or allowing shapes such as triangles that may be more suitable to fill the area desired. Even in programs where a choice of mesh is provided, it is usually recommended for time stepping procedures to provide elements of identical size and shape for the mesh.

Figure 4.3: Illustration of the difficulty of representing a curved surface with square elements.

When carrying out modelling of ultrasonic wave propagation, a chronological simulation is used [150]. The most common is to perform a time step solution such as PZFlex uses. PZFlex uses an explicit time stepping approach, in which the values such as displacements and velocities at any time are used to extrapolate a new set of values a certain time later. This procedure is followed as many times as is necessary until the desired end time is reached. Using this method, the element size and time step size have to be within certain limits to ensure the accuracy and stability of the model. As such the program limits the size and shape of elements in the mesh to be the same to improve solution efficiency and to prevent wave reflections from boundaries in mesh regions with different element sizes. The time step is set to be less than the time taken for the fastest wave to travel the shortest distance between adjacent nodes, which will correspond to the element length. This ensures the stability of the simulation and also ensures that when the convergence test is carried out, it is not only the size of the elements that is changed but also the time step, so that the solution becomes more precise in both distance and time as the number of elements per wavelength is increased.

Despite the restrictions in PZFlex on the shape of the elements, it is a useful tool to use for the propagation of ultrasound due to the ease with which a model

can be established and the reduction in parameters that need to be specified by the user.

Although increasing the mesh density increases the accuracy of the resulting data, it also increases the size of the model as more elements have to be included in the structure. This will lead to increased memory usage and computation times. A balance is therefore required between the number of elements in the system and computational cost of the simulation. If the number elements is too low then the mesh density will not be adequate to represent the sample and so the model will not converge to the desired value. Conversely if the element size is too small, then the simulation will be impractical in terms of memory usage and computation time. A common technique for reducing the number of elements and hence reducing the computational load and time of a finite element simulation is to use knowledge of boundary conditions and symmetry to reduce the size of the model. Symmetry can be used in some cases to allow for half or even a quarter of the overall struc- ture to be representative of the whole model for example, which would reduce the computational time [151].

Absorbing boundary conditions can also be used to limit the computational load by shortening the sample in a direction that isn’t important and removing reflections and scattered signals that are not desired. PZFlex implements the MINT condition (Material Independent Non-reflecting Treatment) that can be found in the literature [151, 152], which makes no assumptions regarding the boundary geometry, angle of incidence of scattered waves or material constitutive properties so that non linear effects can be accommodated.

In the cases considered here, the computational load can be reduced by using the SH wave feature in PZFlex that makes a 2D model with an arbitrary thickness in the 3rd dimension so that a shearing force can be applied in this dimension without having to generate a full representation of this dimension in the model. This allows for simpler models to be generated to determine the behaviour of the waves in the propagation direction circumferentially, but is unable to give any axial information, so 3D models are needed for such an analysis.