Two efficiency-based grid refinement strategies are investigated for adaptive finite element solution of partial differential equations. In each refinement step, the elements are ordered in terms of decreasing local error, and the optimal fraction of elements to be refined is deter- mined based on e±ciency measures that take both error reduction and work into account. The goal is to reach a pre-specified bound on the global error with a minimal amount of work. Two efficiency measures are discussed, 'work times error' and 'accuracy per computational cost'. The resulting refinement strategies are first compared for a one-dimensional model problem that may have a singularity. Modified versions of the efficiency strategies are proposed for the singular case, and the resulting adaptive methods are compared with a threshold-based refinement strategy. Next, the efficiency strategies are applied to the case of hp-refinement for the one-dimensional model problem. The use of the efficiency-based refinement strategies is then explored for problems with spatial dimension greater than one. The work times error strategy is inefficient when the spatial dimension, d, is larger than the finite element order, p, but the accuracy per computational cost strategy provides an efficient refinement mechanism for any combination of d and p.

Submitted towards the Degree of Master of Technology in Mechanical Engineering at the Cape Peninsula University of Technology / A Small Punch Test is a non-destructive technique for evaluating mechanical behaviour. The main advantage of this testing technique is the fact that material can be extraxted from a component in service due to the small dimensions of the speciments.Typical test specimens cut from components are similar in size to a normal human fingernail.

Being able to accurately model soft tissue behaviour, such as that of heart valvular tissue, is essential for developing effective numerical simulations of in-vivo conditions and determining patient-specific care options. Although several analytical material models, based on strain energy functions, have been successful in predicting soft tissue behaviour, complications arise when these models are implemented into finite element (FE) programs due to the incorporation of a penalty parameter for numerically enforcing material incompressibility. Specifically, material parameters determined through non-FE methods may no longer produce a material behaviour that reflects the experimental behaviour once they are used in an FE analysis. Based on commercial finite element software LS-DYNA, an inverse methodology was developed in MATLAB to simultaneously optimize the material parameters and the penalty parameter for the Guccione strain energy model. The methodology produced accurate predictions of the material behaviour under planar equibiaxial testing for five biomaterials used in heart valve cusp replacements.

Contact between beam elements is a specific category of contact problems which was introduced by Wriggers and Zavarise in 1997 for normal contact and later extended by Zavarise and Wriggers to include tangential and frictional contact. In these works, beam elements are assumed to have rigid circular cross-sections and each pair of elements cannot have more than one contact point. The method proposed in the early papers is based on introducing a gap function and calculating the incremental change of that gap function and its variation in terms of incremental change of the nodal displacement vector and its variation. Due to complexity of derivations, specially for tangential contact, it is assumed that beam elements have linear shape functions. Furthermore, moments at the contact point are ignored. In the work presented in this licentiate thesis, we mostly adress the questions of simplicity and robustness of implementations, which become critical once the number of contact is large. In the first paper, we have proposed a robust formulation for normal and tangential contact of beams in 3D space to be used with a penalty stiffness method. This formulation is based on the assumption that contact normal, tangents, and location are constant (independent of displacements) in each iteration, while they are updated between iterations. On the other hand, we have no restrictions on the shape functions of the underlying beam elements. This leads to a mathematically simpler derivation and equations, as the linearization of the variation of the gap function vanishes. The results from this formulation are verified and benchmarked through comparison with the results from the previous algorithms. The proposed method shows better convergence rates allowing for selecting larger loadsteps or broader ranges for penalty stiffness. The performance and robustness of the formulation is demonstrated through numerical examples. In the second paper, we have suggested two alternative methods to handle in-plane rotational contact between beam elements. The first method follows the method of linearizing the variation of gap function, originally proposed by Wriggers and Zavarise. To be able to do the calculations, we have assumed a linear shape function for the underlying beam elements. This method can be used with both penalty stiffness and Lagrange multiplier methods. In the second method, we have followed the same method that we used in our first paper, that is, using the assumption that the contact normal is independent of nodal displacements at each iteration, while it is updated between iterations. This method yields simpler equations and it has no limitations on the shape functions to be used for the beam elements, however, it is limited to penalty stiffness methods. Both methods show comparable convergence rates, performance and stability which is demonstrated through numerical examples. / Kontakt mellan balkelement är en speciell typ av kontaktproblem som först analyserades 1997 av Wriggers och Zavarise med avseende på kontakt i normalriktningen. Teorin utvecklades senare av Zavarise och Wriggers och  inkluderade då även kontakt i tangentiella riktningar. I dessa arbeten antas balkelementen ha ett styvt cirkulärt tvärsnitt och varje elementpar kan inte ha mer än en kontaktpunkt. Metodiken i dessa artiklar bygger på  att en glipfunktion införs och därefter beräknas den inkrementella förändringen av glipfunktionen, och också dess variation, som funktion av den inkrementella förändringen av förskjutningsvektorn och dess variation. På grund av de komplicerade härledningar som resulterar, speciellt för den tangentiella kontakten, antas det att balkelementen har linjära formfunktioner. Dessutom tas ingen hänsyn till de moment som uppstår vid kontaktpunkten. I de arbeten som presenteras i denna licentiatavhandling har vi valt att inrikta oss mot frågeställningar kring enkla och robusta implementeringar, något som blir viktigt först när problemet innefattar ett stort antal kontakter. I den första artikeln i avhandlingen föreslår vi en robust formulering för normal och tangentiell kontakt mellan balkar i en 3D-rymd.Formuleringen bygger på en kostnadsmetod och på antagandet att kontaktens normal- och tangentriktning samt dess läge förblir detsamma (oberoende av förskjutning) under varje iteration. Dock uppdateras dessa storheter mellan varje iteration. Å andra sidan har inga begränsningar införts för formfunktionerna hos de underliggande balkelementen. Detta leder till en matematiskt enklare härledning samt enklare ekvationer, eftersom variationen hos glipfunktionen försvinner. Resultat framtagna med hjälp av denna formulering har verifierats och jämförts med motsvarande resultat givna av andra metoder. Den föreslagna metoden ger snabbare konvergens vilket ger möjlighet att använda större laststeg eller större omfång hos styvheten i kontaktpunkten (s.k. kostnadsstyrhet). Genom att lösa numeriska exempel påvisas prestanda och robusthet hos den föreslagna formuleringen. I den andra artikeln föreslår vi två alternativa metoder för att hantera rotationer i kontaktplanet hos balkelementen. I den första metoden linjäriseras glipfunktionen. Denna metod presenterades först av Wriggers och Zavarise. För att kunna genomföra beräkningarna ansattes linjära formfunktioner för balkelementen. Den här metoden kan användas både med kostnadsmetoder och metoder baserade på Lagrangemultiplikatorer. I den andra föreslagna metoden har vi valt att följa samma tillvägagångsätt som i vår första artikel. Detta betyder att vi antar att kontaktens normalriktning är oberoende av förskjutningarna under en iteration men uppdateras sedan mellan iterationerna. Detta tillvägagångsätt ger enklare ekvationer och har inga begränsningar vad gäller de formfunktioner som används i balkelementen. Dock är metoden begränsad till att utnyttja kostnadsmetoder. Båda de föreslagna metoderna i denna artikel ger jämförbar konvergens, prestanda och stabilitet vilket påvisas genom att lösningar till olika numeriska exempel presenteras. / <p>QC 20160408</p>

An alternative method of conventional prosthesis is  osseointegrated transfemoral implant, in where the prosthesis is fixated directly to the bone. The benefits with this system is increased range of motion, sensory feedback and reduced soft tissue problem. One of the drawbacks of this method is the effect of stress shielding, which could in long term lead to bone loss and bone resorption. The aim of this study is to investigate how the length of the fixture (60, 80 and 100 mm) of OPRA system (Osseointegrated Prosthesis for the Rehabilitation of Amputees) affects the stress distribution in femoral bone and implant during short walk by using Finite Element Methods.  The finite element model used in this study was constructed of three major parts: THUMS model (Total Human Model of Safety) of left thigh, implant and bone graft. The analysis was performed through the software LS-DYNA, with an implicit solver. The loading of the total gait cycle was applied in the distal end of the implant, whereas the proximal end of the thigh was fixed.  The FE simulation revealed lower stress distribution in the distal end of femoral bone, and higher in the proximal end. Implant 60 had lowest effect of stress shielding. The highest stress distribution in OPRA implant was shown in the abutment shaft, in the interface with bone graft. The length of the fixture did not have any impact on the stress distribution in the implant.

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Optimization based domain decomposition methods for the solution of partial differential equations are considered. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains; the constraints are the partial differential equations. First, we consider a linear constraint. The existence of optimal solutions for the optimization problem is shown as is its convergence to the exact solution of the given problem. We then derive an optimality system of partial differential equations from which solutions of the domain decomposition problem may be determined. Finite element approximations to solutions of the optimality system are defined and analyzed as is an eminently parallelizable gradient method for solving the optimality system. The linear constraint minimization problem is also recast as a linear least squares problem and is solved by a conjugate gradient method. The domain decomposition method can be extended to nonlinear problems such as the Navier-Stokes equations. This results from the fact that the objective functional for the minimization problem involves the jump in dependent variables across the interfaces between subdomains. Thus, the method does not require that the partial differential equations themselves be derivable through an extremal problem. An optimality system is derived by applying a Lagrange multiplier rule to a constrained optimization problem. Error estimates for finite element approximations are presented as is a gradient method to solve the optimality system. We also use a Gauss-Newton method to solve the minimization problem with the nonlinear constraint. / Ph. D.

Sensitivity analysis is a useful mathematical tool for many designers, engineers and mathematicians. This work presents a study of sensitivity equation methods for elliptic boundary value problems posed on parameter dependent domains. The current focus of our efforts is the construction of a rigorous mathematical framework for sensitivity analysis and the subsequent development of efficient, accurate algorithms for sensitivity computation. In order to construct the framework, we use the classical theory of partial differential equations along with the method of mappings and the Implicit Function Theorem. Examples are given which illustrate the use of the framework, and some of the shortcomings of the theory are also identified. An overview of some computational methods which make use of the method of mappings is also included. Numerical results for a specific example show that convergence (energy norm) of the sensitivity approximations can be influenced by the specific structure of the computational scheme. / Ph. D.

We study a parameter identification problem for the steady state diffusion equations. In this thesis, we transform this identification problem into a minimization problem by considering an appropriate cost functional and propose a finite element method for the identification of the parameter for the linear and nonlinear partial differential equation. The cost functional involves the classical output least square term, a term approximating the derivative of the piezometric head 𝑢(𝑥), an equation error term plus some regularization terms, which happen to be a norm or a semi-norm of the variables in the cost functional in an appropriate Sobolev space. The existence and uniqueness of the minimizer for the cost functional is proved. Error estimates in a weighted 𝐻⁻¹-norm, 𝐿²-norm and 𝐿¹-norm for the numerical solution are derived. Numerical examples will be given to show features of this numerical method. / Ph. D.

Adaptive mesh refinement schemes are used to find accurate low-dimensional approximating spaces when solving elliptic PDEs with Galerkin finite element methods. For nonlinear PDEs, solving the nonlinear problem with Newton's method requires an initial guess of the solution on a refined space, which can be found by interpolating the solution from a previous refinement. Improving the accuracy of the representation of the converged solution computed on a coarse mesh for use as an initial guess on the refined mesh may reduce the number of Newton iterations required for convergence. In this thesis, we present an algorithm to compute an orthogonal L^2 projection between two dimensional finite element spaces constructed from a triangulation of the domain. Furthermore, we present numerical studies that investigate the efficiency of using this algorithm to solve various nonlinear elliptic boundary value problems. / Master of Science