4.2.2: Discussion on the first order model
4.2.3 : Discussion on the first order model
The results from the initial first order model (i.e. the results shown in table 4. 1 ) are several orders of magnitude greater than the experimental results. These results fail to adequately represent the operation of the Lica and do not explain either the time scale observed in the experimental voltage traces (Le. a 2211S to 3211S in duration overdamped experimental voltage trace, refer to section 3.3.8) or the strange anomaly of the final velocity of the hydrogen plasma (apparent bulk of the plasma) being independent of the positive electrode length (refer to section 3.3.8).
The failings of the first order model can be attributed to its derivation being based upon a filament current. By assuming that the Lorentz force across the plasma is solely the product of this assumption, the effects of the current density and magnetic field distributions have been neglected. If the circuit parameters are held constant and the conductivity of the plasma is either increased or decreased one would expect a noticeable effect on the performance of the accelerator; the first order model is unable to correctly model this effect and other effects such as the velocity skin effect.
The addition of a simple ablation model to the derived first order model demonstrated (as expected) that ablation would have the greatest effect on the performance of the plasma accelerator (as can be seen from the results presented in table 4.2). Unfortunately the addition of the simple ablation model still does not explain the operation of the in fact the simple ablation model suggests the final velocity of the bulk of the hydrogen plasma decreases with an ever increasing electrode length; hence given an infinite acceleration length the simple ablation model would imply that the final velocity of the plasma will tend towards Okmls. This is because the simple ablation model directly incorporates the ablated material into the accelerated plasma mass and neglects any physical interactions this would have on the plasma, its properties and other effects such as plasma restrike (refer to section 1 .3).
Overall it can be said that the first order model is inadequate to accurately model the complex time dependent system of the pulsed coaxial plasma accelerator. The initial first order model described by equations (4. 1 1 ) and (4. 1 2) represents an upper limit to the operation of a coaxial plasma accelerator operating in plasma block mode. These results would be attainable if all the factors effecting the operation of a plasma accelerator (most are mentioned in section 1 .3) could be neglected and the plasma was conductive enough that essentially a filament current would be in effect. Such models can only give accurate answers for simple steady state problems, such as the examples given in [7]. In order to accurately model the operation of the Lica a numerical approach must be used, which will consist of three linked modelling processes: an
4.2.2: Discussion on the first order model
model is necessary to account for the electromagnetic effects i.e. Maxwell ' s equations (refer to section 4.3.2). The magnetohydrodynamic model is required to describe the motional aspects of the plasma i.e. the velocity and compressional effects. And finally, an ion model is necessary to describe the plasma and its interactions i.e. breakdown processes, conductivity calculations, ablation, plasma restrike et cetera.
The remainder of this chapter, section 4.3 in its entirety, is concerned with the development of an initial numerical model to simulate the electromagnetic aspects of the plasma accelerator.
4.3 :Numerical models used to describe the electromagnetic aspects of plasma accelerators and
4.3: Numerical models used to describe the electromagnetic aspects of plasma
accelerators and rail guns
The finite element method (FEM) is the currently accepted approach to solving electromagnetic problems [ 1 05], [ 1 06], [ 1 07]. It is preferred over the previously favoured finite difference (FD)
approach, because of its adaptability in the discretization of complex shaped problems and its far greater accuracy [ 1 08]; an overview of the history and development of FEM in eiectromagnetics can be found in [ 1 09]. Finite difference methods [ 1 1 0], [ I l l ], [ 1 1 2] are very simple to understand and implement compared to FEM; because of this they are still employed in the modelling of plasma accelerators [90], [9 1 ]. Other schemes that are typically used in the solution of fluid flow problems (Navier Stokes) such as the finite volume method (FV or FVM) [ 1 1 3], [ 1 1 4] have also been used to solve the electromagnetic aspects of rail guns [ 1 1 5].
In this thesis FEM was used as the method of choice, because of its general acceptance and the large amount of published literature relating to its implementation. The finite element method is discussed in section 4.3 . 1 , the associated numerical problems (and solutions to these problems) encountered in modelling a coaxial plasma accelerator using this method are covered in sections 4.3.5 to 4.3 . 1 1 . Isoparametric cubic serendipity fmite elements [ 1 08] were used in the numerical program known as MAT AC (that was developed to describe the operation of the refer to section 4.3 . 1 4); these elements give superior solutions in regions containing large gradients compared to lower order elements, as is demonstrated in the test problems presented in section 4.3 . 1 3 .
The use o f edge elements [ 1 05], [ 1 06], [ 1 1 6], [ 1 1 7] (which are a special vector oriented form of isoparametric elements) to model the problem was considered. Edge elements reduce the problem of singularities near re-entrant corners (singularities at re-entrant corners are discussed in section 4.3 .8) [ 1 1 8], [ 1 1 9], [ 1 20] . Edge elements can also be formulated to prevent spurious solutions (spurious solutions are discussed in section 4.3.7); however this is only true for Cartesian coordinates (easily justified by considering the divergence in terms of cylindrical coordinates for the rectangular edge element given in [ 1 06]) and would require the plasma accelerator to be considered in three dimensions (requiring a prohibitive amount of computer memory and time).
The main reason why edge elements were not used is because only tangential boundaries/boundary conditions can be used. If edge elements were applied the problem of the coaxial plasma accelerator, imposing the normal current density boundary condition (refer to section 4.3 .2) would not be possible and instead the tangential component would have to be specified, which is essentially an unknown, on some conducting surfaces.
4.3 . 1 : The finite element method -FEM
4.3.1 : The finite element method -FEM
The finite element method is a numerical technique, which is used to obtain approximate solutions to a set of equations with known boundary conditions. In the finite element method the problem domain is broken up into a series of elements. The elements consist of a series of specified points called nodes or nodal points; the solution to the problem is calculated at these points. The overall structure formed by the elements is referred to as a mesh. There are several finite element schemes: the variational method, the Ritz method and the Galerkin method (also known as the method of weighted residuals) [ 1 08], [ 1 2 1 ]. The Galerkin method is perhaps the easiest of all the methods to apply and for this reason it will be used as the method of choice to model the electromagnetic aspects of the plasma accelerator. In this approach the equation (to be integrated over the computational domain) is multiplied by a trial function (also referred to as an interpolation, test or shape function). The trial function is taken to be equal to the sought after solution to the problem and is typically represented by a polynomial function. The trial function is required to fulfill two conditions. Firstly, the order of the polynomial function chosen must be equal to the degrees of freedom of the element; in other words the polynomial must contain enough terms to define a set of equations in terms of the sought after solution for all the nodes of an element. Secondly, the continuity of the trial function must be the same as the equation being solved. This point is discussed in a practical approach in [1 08J and in a theoretical sense in [ 1 22]. In simple terms, if a polynomial trial function is used it will have what is known as Co continuity (the solution is guaranteed to be continuous across an element
surface); hence it will be sufficient to give a unique variation of the derivative of the solution across the element surface. The highest order derivative that can be solved with Co continuity is a first order derivative. Due to this second requirement if a polynomial trial function is used then the equations to be solved must be simplified using integration by parts and the vector relationships in appendix 4A to derive derivative terms no higher than first order. Providing these requirements have been fulfilled the resulting equations (simpli fied by integration by parts if necessary) are integrated with respect to the computational volume and the resulting matrix is solved using the known boundary conditions.
The following example using a linear trial function across a one dimensional problem domain will be used to demonstrate the method by integrating equation (4.20) in the range of 0 < x < 1 .
(4.20)
4.3. 1 : The finite element method -FEM Subj ect to the boundary conditions
U(o)= 1
u(1) = e1
F irst equation (4.20) is rearranged, then multiplied by the trial function
w
(4.2 1 ) (4.22)
(4.23)
Integration by parts on the above equation is used to produce what is known as a weak formulation or weak form of the problem: