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Indiana University-Purdue University Indianapolis (IUPUI)

In this dissertation we are concerned with the analysis of the finite element method for the time-dependent Maxwell interface problem when Nedelec and Raviart-Thomas finite elements are employed and preconditioning of the resulting linear system when implicit time schemes are used. We first investigate the finite element method proposed by Makridakis and Monk in 1995. After studying the regularity of the solution to time dependent Maxwell's problem and providing approximation estimates for the Fortin operator, we are able to give the optimal error estimate for the semi-discrete scheme for Maxwell's equations. Then we study preconditioners for linear systems arising in the finite element method for time-dependent Maxwell's equations using implicit time-stepping. Such linear systems are usually very large but sparse and can only be solved iteratively. We consider overlapping Schwarz methods and multigrid methods and extend some existing theoretical convergence results. For overlapping Schwarz methods, we provide numerical experiments to confirm the theoretical analysis.

Nucleation and propagation of cracks under fretting conditions has been a subject of study for many years. An extensive experimental investigation to study these cracks was undertaken by Royal Aerospace Establishment (RAE Farnborough). Of particular interest to RAE was an Aluminium alloy (L65) developed for aerospace applications. Many researchers have studied fretting damage and fatigue cracks. Some have examined damage development due to wear, whilst others have analysed cracks under linear elastic fracture mechanics (LEFM) domain. To date, no attempt has been made to develop an integrated numerical model which incorporates all aspects of fretting fatigue i.e. nucleation, initial (or early) crack growth, and long crack propagation. The development of such a model is the principal aim of this work. It is expected that the integrated approach will provide the basis for a standard fretting fatigue analysis of other materials, components, and structures using the finite element method (FEM).This study uses the earlier experimental results with RAE as the reference for comparison. The approach followed is to implement the various stages of fretting in a commercial finite element code, ABAQUS. Unlike previously used simple FE models, both specimen (Aluminium alloy) and the fretting pad (steel) are modelled to simulate the real contact conditions including slip. Various predictive models for crack nucleation (based on damage) and propagation (based on fracture mechanics) are developed, tested, and implemented in ABAQUS. Results clearly show that these models together provide a good estimation tool for predicting total life in complete contact fretting fatigue. It is envisaged that the integrated model will be easily utilised for other materials, components, and structures subjected to fretting fatigue conditions with minimum experimental testing required.

In this dissertation we develop, analyze and implement effective numerical methods for multiscale phenomena arising from flows in heterogeneous porous media. The main purpose is to develop innovative numerical and analytical methods that can capture the effect of small scales on the large scales without resolving the small scale details on a coarse computational grid. This research activity is strongly motivated by many important practical applications arising in contaminant transport in heterogeneous porous media, oil reservoir simulations and subsurface characterization. In the work, we investigate three main multiscale numerical methods, i.e., multiscale finite element method, partition of unity method and mixed multiscale finite element method. These methods employ limited single or multiple global information. We apply these numerical methods to partial differential equations (elliptic, parabolic and wave equations) with continuum scales. To compute the solution of partial differential equations on a coarse grid, we define global fields such that the solution smoothly depends on these fields. The global fields typically contain non-local information required for achieving a convergence independent of small scales. We present a rigorous analysis and show that the proposed global multiscale numerical methods converge independent of small scales. In particular, a global mixed multiscale finite element method is extensively studied and applied to two-phase flows. We present some numerical results for two-phase simulations on coarse grids. The numerical results demonstrate that the global multiscale numerical methods achieve high accuracy.

Classical alternating direction (AD) methods for parabolic equations, based on some standard implicit time stepping procedure such as Crank-Nicolson, can have errors associated with the AD perturbations that are much larger than the errors associated with the underlying time stepping procedure . We plan to show that minor modifications in the AD procedures can virtually eliminate the perturbation errors at an minor additional computational cost. A mixed finite element method is applied in the spactial variables. Similar to the finite difference and finite element methods in spactial variables, we plan to have the same accuracy in time. A convergence analysis can also be shown .

Structural engineers have used finite element methods for the design of reinforced concrete plate type structures for decades. The theory behind this method is well researched, however, there is still a lack of direction on how to use the information obtained from this type of analysis to practically design reinforced concrete structures for strength and serviceability criteria. The literature study reviews the analysis of concrete plate type structures using traditional and finite element methods and highlights the difference between linear and non-linear finite element analysis. It is apparent that when designing and detailing using a FE analysis, a great deal is left up to engineering judgement, especially in areas of the structures where peak load effects (singularities) are experienced. In this thesis these peak areas are investigated, in an effort to provide insight into the actual behaviour of the structure as opposed to the theoretical results obtained from a FE analysis. The research consists of both numerical, (linear and non-linear FE analyses) and practical experimental work performed on different types of concrete plate type structures, including concrete pad foundations and simply supported flat slabs. The response to loading, i.e: cracking characteristics, softening of the concrete, moment redistribution, variation of the strain in reinforcement across the section, and deflection is observed and discussed. The results show that the traditional simplified methods are adequate with respect to overall strength. Finite element peaks or singularities may be averaged or smoothed without compromising durability and serviceability. Suggestions on how the reinforcement obtained from linear finite element methods be detailed are given. / Dissertation (MEng)--University of Pretoria, 2013. / gm2014 / Civil Engineering / unrestricted

Thesis (PhD)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Dunnage bags are an inflatable dunnage variant, positioned and inflated between goods in multi-modal containers to restrain and protect the goods while in transit. This project endeavours to develop a simple method of generating new numerical prototypes for dunnage bags suitable for simulating operational loading of the bags. Previous research has produced a model that simulates the inflation of a paper dunnage bag using a simple pressure load. A dunnage bag reinforced with plain-woven polypropylene was chosen as the test case. Woven polypropylene is a highly non-linear, non-continuous, non-homogeneous material that requires specialised material models to simulate. A key aspect of this project was to develop a simple method for characterising woven-polypropylene and replicating it's response with material models native to LS-DYNA. The mechanical response of the plain-woven polypropylene was tested using a bi-axial tensile test device. The material response from physical testing was then mapped to two material models using the numerical optimiser LS-OPT. The response of the calibrated material models was found to correlate well with the measured response of the woven material. Dunnage bags are subjected to cyclic loading in operation. In order to capture the effects of compressing the contained gas, a gas inflation model was added to the model that calculates the pressure in the bag based on the Ideal Gas Law. A full bag model making use of the calibrated material models and the inflation model was subjected to a cycled boundary condition simulating loading and unloading of an inflated dunnage bag. The two prototype models captured the pressure drop in the bag due to material plastic deformation and the restraining force produced by the bag to within 10 %. The prototype models were also found suitable for predicting burst pressure in voids of arbitrary size and shape. / AFRIKAANSE OPSOMMING: Stusakke is 'n opblaasbare soort stumateriaal wat tussen goedere in multimodale vraghouers geposisioneer en opgeblaas word om sodoende die goedere vas te druk en te beskerm tydens vervoer. Hierdie projek poog om 'n eenvoudige manier te ontwikkel om nuwe numeriese prototipes vir stusakke, geskik om operasionele lading van die sakke te simuleer, te ontwikkel. Vorige navorsing het 'n model ontwikkel wat die opblaas van 'n papier stusak met eenvoudige drukkrag simlueer. 'n Hoë-vlak stusak versterk met plein-geweefde polipropileen, is gekies om getoets te word. Geweefde polipropoleen is 'n hoogs nie-lineêre, onderbroke, nie-homogene materiaal wat gespesialiseerde materiaalmodelle nodig het vir simulasie. Een van die fokuspunte van hierdie projek is om 'n eenvoudige metode te ontwikkel om die karaktereienskappe van polipropoleen te identifiseer en die gedrag daarvan na te maak met die materiaalmodelle van LSDYNA. Die meganiese reaksie van die plein-geweefde polipropoleen is getoets met 'n biaksiale/tweeassige trektoets-toestel. Die materiaal se reaksie op die fisiese toets is ingevoer op 'n numeriese optimiseerder, LS-OPT, om op die materiaalmodelle te toets. Die reaksie van die gekalibreerde materiaalmodelle het goed gekorelleer met die gemete reaksie van die geweefde materiaal. Stusakke word tydens diens onderwerp aan sikliese lading. Om die effek van die saamgepersde gas vas te stel is 'n gas-opblaasbare model bygevoeg by die model wat die druk in die sak bereken, soos gebaseer op die Ideale Gas Wet. 'n Volskaalse sakmodel wat gebruik maak van die gekalibreerde materiaalmodelle en die opblaas-model is onderwerp aan sikliese grensvoorwaardes wat die lading en ontlading van 'n opblaasbare stusak simuleer. Die twee prototipe modelle het die drukverlies in die sak a.g.v. die materiaal-plastiek vervorming en die bedwingingskrag van die sak beperk tot 10 %. Die protoyipe modelle is ook geskik bevind om barsdruk in arbitrêre leemtes te voorspel.

This thesis reports onreview and research work carried out on the numerical analysis of elastomers. The two numerical techniques investigated for this purpose are the finite and boundary element methods. The finite element method is studied so that existing theory is used to develop a finite element code both to review the finite element method as applied to the stress analysis of elastomers and to provide a comparison of results and numerical approach with the boundary element method. The research work supported on in this thesis covers the application of the boundary element method to the stress analysis of elastomers. To this end a simplified regularization approach is discussed for the removal of strong and hypersingularities generated in the system on non-linear boundary integral equations. The necessary programming details for the implementation of the boundary element method are discussed based on the code developed for this research. Both the finite and boundary element codes developed for this research use the Mooney-Rivlin material model as the strain energy based constitutive stress strain function. For validation purposes four test cases are investigated. These are the uni-axial patch test, pressurized thick wall cylinder, centrifugal loading of a rotating disk and the J-Integral evaluation for a centrally cracked plate. For the patch test and pressurized cylinder, both plane stress and strain have been investigated. For the centrifugal loading and centrally cracked plate test cases only plane stress has been investigated. For each test case the equivalent results for an equivalent FEM program mesh have been presented. The test results included in this thesis prove that the FE and BE derivations detailed in this work are correct. Specifically the simplified domain integral singular and hyper-singular regularization approach was shown to lead to accurate results for the test cases detailed. Various algorithm findings specific to the BEM implementation of the theory are also discussed.

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.