In science and engineering, many simulations are carried out over domains consisting of multiple materials separated by curves/surfaces. If partial differential equations (PDEs) are used to model these simulations, it usually leads to the so-called interface problems of PDEs whose coefficients are discontinuous. In this dissertation, we consider nonconforming immersed "nite element (IFE) methods and error analysis for interface problems. We "first consider the second order elliptic interface problem with a discontinuous diffusion coefficient. We propose new IFE spaces based on the nonconforming rotated Q1 "finite elements on Cartesian meshes. The degrees of freedom of these IFE spaces are determined by midpoint values or average integral values on edges. We investigate fundamental properties of these IFE spaces, such as unisolvency and partition of unity, and extend well-known trace inequalities and inverse inequalities to these IFE functions. Through interpolation error analysis, we prove that these IFE spaces have optimal approximation capabilities. We use these IFE spaces to develop partially penalized Galerkin (PPG) IFE schemes whose bilinear forms contain penalty terms over interface edges. Error estimation is carried out for these IFE schemes. We prove that the PPG schemes with IFE spaces based on integral-value degrees of freedom have the optimal convergence in an energy norm. Following a similar approach, we prove that the interior penalty discontinuous Galerkin schemes based on these IFE functions also have the optimal convergence. However, for the PPG schemes based on midpoint-value degrees of freedom, we prove that they have at least a sub-optimal convergence. Numerical experiments are provided to demonstrate features of these IFE methods and compare them with other related numerical schemes. We extend nonconforming IFE schemes to the planar elasticity interface problem with discontinuous Lam"e parameters. Vector-valued nonconforming rotated Q1 IFE functions with integral-value degrees of freedom are unisolvent with appropriate interface jump conditions. More importantly, the Galerkin IFE scheme using these vector-valued nonconforming rotated Q1 IFE functions are "locking-free" for nearly incompressible elastic materials. In the last part of this dissertation, we consider potential applications of IFE methods to time dependent PDEs with moving interfaces. Using IFE functions in the discretization in space enables the applicability of the method of lines. Crank-Nicolson type fully discrete schemes are also developed as alternative approaches for solving moving interface problems. / Ph. D.

Knowledge about the dynamic behaviour is a basic condition for a secure operation of modern machine tools. Hence numerical methods predicting the dynamic properties are gaining in importance. Usually for complex and coupled structures, the results of dynamic property calculation are yet insufficient. Therefore Finite Element model updating is a tool to improve the hypothetical factor of the analysis. Within the present thesis Finite Element modelling is performed using the example of the “Scherenkinematik”, a machine tool based on hybrid-kinematics. Initially the results of an Experimental Modal Analysis are evaluated by identifying Modal parameters and deriving possible structural modifications. In the second part of the thesis, the machines Finite Element model is created using the FEA-Software ANSYS. Afterwards the Finite Element model updating is performed by coupling ANSYS and the CAE-Software FEMtools. Therefore two approaches are formulated and tracked. It turns out, that there is no improvement of the analytical and experimental models correlation, neighter with nor without a steady reduction of the search domain needed for mode coupling. It is reasoned, that the characteristics and the results of an Finite Element updating process are affected by the quality of the model at start time and the approach as well as the technique chosen for model updating and parameter modification. Therefore the CAE-Software FEMtools is suitable to only a limited extent for Finite Element updating of strongly coupled mechanical structures as a result of the sensitivity analysis used for parameter modification.

Indiana University-Purdue University Indianapolis (IUPUI) / Advances in peripheral nerve electrode technology have outpaced the advances in chronic implantation reliability of the electrodes. An observable trend is the increased deposition of fibrotic encapsulation tissue around the electrode to shift its position away from the implantation site and subsequently reducing performance. A finite element model (FEM) is developed in conjunction with tensile testing and digital image correlation of strain to understand the relationship between cuff electrode attachment and the strain environment of the nerve. A laminar and bulk nerve model are both developed with material properties found in literature and geometry found from performing histology. The introduction of a cuff electrode to an axially stretched nerve indicates a significant behavior deviation from the expected response of the axial strain environment. When implemented in ex-vivo tensile testing, results indicate that the reduction of strain is statistically significant but becomes much more apparent when paired with a digital image correlation system to compare predicted and measured effects. A robust FEM is developed and tested to emphasize the effect that the boundary conditions and attachment methodology significantly effects the strain environment. By coupling digital image correlation with FEM, predictive models can be made to the strain environment to better design around the long term chronic health of the implant.

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Femoroacetabular impingement (FAI) is recognized as a pathomechanical process that leads to hip osteoarthritis (OA). It is hypothesized that mechanical stimuli are prominent at higher range of motions in hips with cam FAI (aspherical femoral head-neck deformity). Adverse loading conditions can impose elevated mechanical stimuli levels at the articulating surfaces and underlying subchondral bone, which plays a predominant mechanical role in early OA. The aim of this research was to determine the levels of mechanical stimuli within the hip, examining the effects of severe cam impingement on the onset of OA, using patient-specific biomechanics data, CT data, and finite element analysis (FEA). Patient-specific hip joint reaction forces were applied to two symptomatic patient models and two control-matched models, segmented from patient-specific CT data. The finite element models were simulated to compare the locations and magnitudes of mechanical stimuli during two quasi-static positions from standing to squatting. Maximum-shear stress (MSS) was analyzed to determine the adverse loading conditions within the joint and strain energy density (SED) was determined to examine its effect on the initiation of bone remodelling. The results revealed that peak mechanical stimuli concentrations were found on the antero-superior acetabulum during the squatting position, underlying to the cartilage. The MSS magnitudes were significantly higher and concentrated for the FAI patients (15.145 ± 1.715 MPa) in comparison with the MSS magnitudes for the control subjects (4.445 ± 0.085 MPa). The FAI group demonstrated a slight increase in peak SED values on the acetabulum from standing (1.005 ± 0.076 kPa) to squatting (1.018 ± 0.082 kPa). Insignificant changes in SED values were noticed for the control subjects. Squatting orients the femoral head into the antero-superior acetabulum, increasing the contact area with the cartilage and labral regions, thus resulting in higher peaks behind the cartilage on the acetabulum. The resultant location of the peak MSS and SED concentrations correspond well with the region of initial cartilage degradation and early OA observed during open surgical dislocation. Due to the relatively low elastic modulus of the articular cartilage, loads are transferred and amplified to the subchondral bone. This further suggests that elevated stimuli levels can provoke stiffening of the underlying subchondral plate, through bone remodelling, and consequently accelerating the onset of cartilage degradation. Since mechanical stimuli results are unique to their patient-specific loading parameters and conditions, it would be difficult to determine a patient-specific threshold to provoke bone remodeling at this stage.

Through the last decades several nonlocal models of linear elasticity have been introduced as enhancements of the Cauchy-elastic model, often with the purpose of providing an improved mechanical description of solids at the microscale level. Although many efforts have been devoted to the analytical formulation of these advanced constitutive models, a definitive interpretation of the relevant static quantities is still incomplete and Finite Element (FE) solvers are practically unavailable. In this thesis, after providing a mechanical interpretation to the static quantities involved in strain gradient (of Mindlin type) elastic materials, an overview on the possible quadrilateral Hermitian finite elements is given to treat quasi-static plane problems. Beside the classical finite elements inspired by those adopted for modeling Kirchhoff plates, an alternative quadrilateral self-constrained finite element formulated through Lagrange multipliers is also proposed. With reference to a hexagonal lattice structure, for which the equivalent constitutive tensors have been recently derived as closed-form expressions, the developed FE codes are exploited to assess the reliability of modelling lattices through higher-order constitutive equations. These analyses are developed for one-dimensional and two dimensional problems, where the former are considered for both homogeneous layers (with a finite size in one direction) and rod-type structures (with a finite uniform cross section along one direction). It is confirmed that higher-order modelling improves the mechanical description. In particular, the macroscale response is shown to be strongly affected by higher-order contributions in the presence of extreme elastic contrast between microstructural elements. Indeed, in this last case, only higher-order modeling captures a non-null residual stiffness, which vanishes in the framework of classical models. Therefore, higher-order modeling becomes important not only to describe the mechanical response at a microlevel, but also for macrolevel modelling, when extreme mechanical properties are addressed. The presented results pave the way to a refined modelling of architected materials leading to improved design of microstructures displaying innovative mechanical features.

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Experimental Spatial Dynamics Modeling (ESDM) is the new process of constructing a three dimensional, complex-valued dynamic model of a harmonically vibrating structure using numerical models and laser-based experimental data obtained from a Scanning Laser Doppler Vibrometer (SLDV). In ESDM process, a finite element formulation is used to construct a numerical model of a structure. A conventional finite element such as rod, beam, or plate element, can be used to construct the numerical model of a structure from its mid-plane. In this research, the damped flat shell element is developed to construct the numerical models of a cantilever beam and a simply supported flat plate. The velocity transformation function developed in this research will make possible to use the FE model, constructed by the damped flat shell element, and the laser-based experimental data within a framework of ESDM in the consistent manner. / Master of Science

In this thesis, we develop the a new immersed finite element(IFE) space formed by piecewise linear polynomials defined on sub-elements cut by the actual interface curve for solving elliptic interface problems on interface independent meshes. A group of geometric identities and estimates on interface elements are derived. Based on these geometric identities and estimates, we establish a multi-point Taylor expansion of the true solutions and show the estimates for the second order terms in the expansion. Then, we construct the local IFE spaces by imposing the weak jump conditions and nodal value conditions on the piecewise polynomials. The unisolvence of the IFE shape functions is proven by the invertibility of the well-known Sherman-Morrison system. Furthermore we derive a group of fundamental identities about the IFE shape functions, which show that the two polynomial components in an IFE shape function are highly related. Finally we employ these fundamental identities and the multi-point Taylor expansion to derive the estimates for IFE interpolation errors in L2 and semi-H1 norms. / Master of Science