Modificación de la resolución de concesión

The main body of this thesis consists of 6 chapters. Each of the chapters will consist of a similar form; the chapter will begin with a review of the relevant literature which will contextualise and explain the motivations for each of the chapters, this will be followed by a theory section, and then where applicable numerical examples. A more specific breakdown of the information contained within each chapter is detailed below.

Chapter 2: Discontinuous Galerkin methods

Chapter 2 begins with an overview of numerical methods for the spatial discretisation of PDEs, in particular presenting the history and development of DG and Interior Penalty (IP) methods since their inception in the 1970’s. Next the preliminary mathematical nomenclature for DG methods is presented, which is requisite for the discussions which will follow for the remainder of the thesis. After this the literature related to DG methods as applied to the specific classes of equations found in this thesis, first-order advection and second-order quasilinear diffusion, are reviewed with attention paid to error analysis where it has been performed. Furthermore, a generic problem belonging to each of these classes is discretised using the relevant DG variant and the discretisation explained.

Chapter 3: Level set method

Chapter 3 introduces and presents the preliminary mathematical nomenclature for the level set method. In particular three aspects of the level set method are outlined; the formulation of a generic level set evolution problem; level set initialisation and reinitialisation; and narrow banded level set methods. Also presented in this chapter is a literature review concerning ex- isting high-order discretisations of the level set evolution equation with a focus on DG methods where they have been applied.

Chapter 4: Level set reinitialisation

Chapter 4 presents the research completed in the area of level set reinitialisation. This begins with a literature review of the existing methods of level set reinitialisation, again focussing on DG discretisations wherever they occur. This is followed by the proposal of a novel reinitialisation method referred to as the Elliptic Reinitialisation method, the development of which is explained in detail and which is shown through experimentation to be of significantly higher accuracy than is capable with competitive methods. A second novelty is presented whereby through reformula- tion of the Elliptic Reinitialisation method, a Parabolic Reinitialisation method is also proposed. The two developed reinitialisation methods are compared and whilst both methods display sim- ilar levels of accuracy, the stability of the parabolic formulation is dependent on a problem dependent time step and as such the Elliptic Reinitialisation is decided to be the preferable method. Also presented in this chapter are discussions concerning the enforcement of boundary

conditions on immersed implicit level set interfaces, and the necessity of narrow banding for certain applications of the level set method.

Chapter 5: Level set evolution

Chapter 5 proposes a DG methodology for solving level set evolution problems. In particular the level set equation is simplified, which is made possible by frequent and accurate level set reinitial- isation, and then discretised using a DG method using a novel flux. Narrow banding is discussed and with it a novel technique is presented for extrapolating the level set function to elements outside of the narrow band, which is required as the narrow band itself evolves to follow the evolving interface. Anderson acceleration is discussed as a method for improving the convergence of the fixed point iterative method used to solve the reinitialisation and extrapolation problems. The constituents of the methodology; the evolution, reinitialisation, and extrapolation equations and their associated technology are combined and used to solve a number of numerical examples. Chapter 6: Level set method: adaptive mesh refinement

Chapter 6 proposes a novel strategy and criterion for hp-adaptive mesh refinement in the con- text of the DG discretised narrow banded level set methodology presented in Chapter 5. The chapter begins with a review of the literature concerning existing refinement strategies for level set methods. Then after presenting the proposed refinement strategy, a number of level set reini- tialisation, and level set evolution example problems are solved this time on hp-adaptive meshes. Chapter 7: Topology Optimisation

Chapter 7 presents a brief foray into topology optimisation using the DG discretised, hp- adaptive, narrow banded level set methodology formulated in the preceding chapters. A lit- erature review is presented discussing a history of the methods used for topology optimisation, and thus where the level set methodology presented thus far would fit in such a context. The requisite theory for using a shape sensitivity approach to solving a minimum compliance problem for linear elastic structures with a constraint on the maximum allowed amount of material is presented which is combined with the proposed level set methodology before solving an example problem involving the design of a cantilever beam under an applied traction.

Chapter 8: Conclusions

Chapter 8 presents a summary of the ideas presented in the thesis, and highlights in particular the novel developments made during the research period. The chapter then proceeds to discuss issues encountered which are yet to be overcome, as well as other areas of interest which were ultimately beyond the scope of the research presented in this thesis given the time constraints of the research period, as parts of a larger discussion on suggested areas of future work.

Whilst mathematical notation will be introduced as and when necessary throughout the thesis, stated here are a number of conventions which will be consistent throughout the presented nomenclature. Vector valued functions and variables always use bold-face notation, for example,

the spatial variable can be denoted x = {x, y}. Subscripts are used extensively as identifiers between variables and functions which denote the same general meaning but differ in specifics,

for example, E_L2 and E_DG both denote errors with the subscript identifying the specific norm

in which the error is computed. In the case that a variable needs both an identifier and an index as a subscript, these are separated by a comma in the order, identifier then index. It is noted that the comma subscript is not used in this thesis, as is sometimes found in the literature, to denote a partial derivative. Indices denoting time step or iteration are always superscripts, and use the symbol m, n or k. In the case of nested iterative methods, multiple superscripts will be used, again separated by a comma. Where algorithms are presented, functions and variables are often given descriptive names as opposed to symbols, in such a case these names are written in camel case and use the Latin Modern Typewriter font, for example, errorHandle is a variable containing a handle which points to the appropriate error to be computed. Furthermore, where names are used instead of symbols, operations such as multiplication will always be stated explicitly to avoid confusion.

Chapter 2

Discontinuous Galerkin Methods

2.1

Overview

Finite Element (FE) analysis is an almost ubiquitous numerical tool for approximating the solution to problems in engineering. The standard (i.e. most popular) FE method is known as the Continuous Galerkin (CG) FE method. For a problem with a solution which belongs to a function space, V , the key component of a Galerkin FE method is to look for a solution in a

finite dimensional subspace of that space, Vh (where the subscript h refers to a discretisation

parameter, often and in this case related to the size of the elements used to discretise the domain on which the problem is to be solved). The classical conforming CGFE method then,

takes its name from the restrictions imposed on this approximation space, Vh, which are; that it

is conforming i.e. Vh ⊂ V , and in particular that the functions which form a basis of the space

are continuous throughout the problem domain (and therefore are continuous across element boundaries).

In 1973, Reed and Hill [35], designed an explicit non-conforming numerical method for solving the neutron transport problem in which there was no requirement on interelement continuity

on the approximation space, Vh, for the first time in a finite element context. In their paper,

Reed and Hill presented a piecewise continuous method where the problem is formulated locally on each element in the problem domain, and a flux term is used to pass information between adjacent elements. This was compared against a strictly continuous formulation, where it was demonstrated by experiment that the discontinuous method was superior both in terms of ac- curacy and stability. This was a significant result as traditionally FE methods had performed poorly when attempting to solve first-order hyperbolic Partial Differential Equations (PDEs), compared with other numerical methods of the time, and therefore had typically been ignored by the wider community for use in this context. In 1974, LeSaint and Raviart noted this sig- nificance, and produced an analysis of Reed and Hill’s method [36] in which they showed that Reed and Hill’s method was a generalisation of the CGFE method, and subsequently named Reed and Hill’s method, the Discontinuous Galerkin (DG) method.

The original DG scheme of Reed and Hill, for solving a transport problem, was influenced by earlier works on solving transport problems, whereby, as it is known that advection is a directional phenomenon, the appropriate method of passing information between elements is

to make use of an upwind type flux. Diffusion however is a non-directional phenomenon, and as such non-conforming FE methods for diffusion type problems might therefore be built in a similar way to Reed and Hill’s method, except in this case by passing information across element faces using a central/average type flux. Around the same time that Reed and Hill published their article the methods of Nitsche [37] and Babuˇska [38] which allowed one to weakly impose Dirichlet boundary conditions, were being extended, much in this vein, to allow for analogous methods to be used to enforce interelement continuity. These methods are known as the Interior Penalty (IP) methods. Whilst it is slightly more difficult to pinpoint an original IP method,

one early example was presented in the 1973 paper by Babu˘ska and Zl´amal [39], in which a

penalty type method was used to weakly impose C1 _{continuity for the fourth-order biharmonic}

equation. Other methods which used an approach more analogous to Nitsche’s method includes

the 1977 article by Baker [40], which again imposed C1 _{continuity on C}0 _{elements for fourth-}

order problems, the 1976 work by Douglas and Dupont [41] which penalised the jump in the normal derivative to enforce continuity for second-order elliptic and parabolic PDEs, the 1978 work by Wheeler [42] which included generalisations of the consistency, symmetry and penalty terms of Nitsche’s method across element edges to solve elliptic PDEs, and the 1979 PhD Thesis of Arnold [43] which presented and analysed a similar method to that of Wheeler for nonlinear elliptic and parabolic PDEs. The idea however, that IP methods were DG methods using a different type of flux wasn’t noticed until the 1990’s and as such the development of DG methods and IP methods continued independently in the interim.

After the initial attempts at using DG and IP methods in the 1970’s, further developments were relatively sparse in the following decade. One active area of research at the time however, was methods for solving nonlinear hyperbolic conservation laws. In 1982, Chavent and Solzano [44] published an article which attempted to extend the works of of Reed and Hill and LeSaint and Raviart to solve problems of this type, however, one of the main issues with DG methods during this period, was the time discretisation. An implicit solver would require an expensive global nonlinear solve, whereas the first-order explicit Euler method which was used by Chavent and Solzano [44] suffers from a severe time step restriction. In the case of IP methods, Arnold [45] suggests that the reason for the lack of progress may have been to do with the difficulty in finding optimal penalty parameters, as well as the methods never being proven to have significant advantages over a classical CGFE approach.

In the same two decades however, largely driven by the desire to compute accurate solutions to nonlinear hyperbolic systems, there was a vast amount of research completed in the area of high resolution Finite Difference (FD) and Finite Volume (FV) schemes. An important starting point in this context was Godunov’s method [46], which was a first-order finite volume scheme, upon which many of the high resolution schemes were based. The main novel idea in Godunov’s method was to update the solution at each time step by solving exactly a Riemann problem at each cell interface and averaging the solution to these Riemann problems over the domain. In this way Godunov had extended the first-order upwind scheme of Courant, Isaacson and Rees [47] to nonlinear systems of hyperbolic conservation equations. One of the main issues with first- order methods such as Godunov’s method is that they tend to be very diffusive, and therefore

cause discontinuities, which might be physical, to smooth out over time. Higher-order methods such as the Lax-Wendroff scheme [48], however, whilst providing higher resolution in smooth parts of the solution introduce spurious oscillations near to local extrema and particularly in the regions where a solution is discontinuous. In fact Godunov in his article, [46], had proven that higher-order methods, could not both preserve monotonicity (i.e. not introduce oscillations) and also be higher than first-order accurate. The main issue to overcome then was the question of how to resolve discontinuities in a solution over relatively few cells, whilst providing high- order accuracy in smooth regions without introducing erroneous oscillations. Methods presented which aimed to overcome this issue began to be published in the 1970’s, with the series of articles by Van Leer [49–53]. In these articles Van Leer introduced the Monotone Upstream- Centered Scheme for Conservation Laws (MUSCL) and with it the idea of flux/slope limiting, by which Godunov’s scheme was extended to include adaptive higher-order (in this case linear) approximations to the solution at cell interfaces. The idea was that the higher-order fluxes would provide a higher resolution where the solution was smooth, and the flux could then be limited to prevent oscillations in areas where the solution was sharp. In this way the method, and later higher-order variants for example [54–56], were able to ensure that the total variation (a measure of the oscillation in a solution) was non-increasing (also known as Total Variation Diminishing (TVD)). A related advancement in this respect was introduced in 1981 by Roe [57], who noticed that much of the information gained by solving the set of exact Riemann problems was lost after the solution at cell interfaces was then averaged over the domain. Given that these Riemann problems, especially in the case of nonlinear systems of equations, could be expensive to solve, Roe considered that it may be possible to obtain good results by replacing the Riemann problem with a cheaper to compute approximation. In that initial paper by Roe and similar papers which followed, for example [58, 59], this was achieved by either approximating the Riemann states and then computing the physical flux, or by approximating directly a numerical flux. In 1987, it was shown [60] that schemes which were total variation non-increasing were at most first-order accurate due to the degenerated accuracy in the regions of local extrema (which may be smooth). The solution to this issue was the development of a suitable relaxation of the the TVD property. One approach in this vein, presented in a series of articles by Harten et al. [60, 61], replaced the TVD property by the more lenient Uniformly Non-Oscillatory (UNO) property which allowed the total variation to increase, but maintained that the number of local extrema must be non-increasing, and then eventually by the even more lenient Essentially Non- Oscillatory (ENO) property which allowed for the number of local extrema to increase as well as the total variation, however the increase in the total variation was bounded to be on the order of the cell size, h. A different approach developed by Shu [62], was to replace the TVD property with the Total Variation Bounded (TVB) property by which the total variation is bounded by a positive constant for all time. The schemes developed under the UNO, ENO and TVB frameworks were designed to use arbitrarily high-order interpolation spaces and therefore be able to maintain higher-order accuracy in smooth regions. However, as is noted in [63], high resolution finite volume schemes are not in general formally high-order accurate, in that even on smooth parts of a solution there can be a degradation of accuracy as a result of the coupling

between characteristic components for all but the simplest linear advection problems.

Despite the lack of research directly into DG schemes, by virtue of the discontinuous finite element spaces which can be used in the DG paradigm, information is passed between cells by defining a numerical flux, in much the same way as finite volume methods, and therefore the research on high resolution schemes was able to be naturally incorporated into the finite element framework through DG methods. Thus in the late 1980’s the works of Van Leer on slope limiters [52] and Harten on TVD schemes [60] was able to be utilised in the context of DG allowing for the series of articles by Cockburn and Shu [64–68] which presented the Runge-Kutta (RK) DG method for multidimensional systems of nonlinear hyperbolic problems. Furthermore, by the 1990’s with the developments in the area of DG methods for problems with non-negligible diffusive parts by authors such as Baumann and Oden [69] and Bassi and Rebay [70], the similarity between IP and DG methods was finally noticed, which then prompted a resurgence of research into IP methods for diffusive PDEs with aims to exploit what was becoming an apparent abundance of advantages of discretising problems using DG methods. Some of the advantages of DG discretisations over both traditional FE schemes and also the aforementioned high resolution FV schemes include; the methods’ formal high-order accuracy, their high level of parallelisability, their ability to easily incorporate hp-adaptivity, their ability to deal with complex geometries, their nonlinear stability and their ability to deal with discontinuous solutions. For these reasons, this thesis proceeds in a similar fashion by attempting to extend the domain of DG discretisations further, into the context of level set based topology optimisation methods.