# Browsing by Subject "Finite element method"

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Item Open Access A Hybrid Spectral-Element / Finite-Element Time-Domain Method for Multiscale Electromagnetic Simulations(2010) Chen, JiefuIn this study we propose a fast hybrid spectral-element time-domain (SETD) / finite-element time-domain (FETD) method for transient analysis of multiscale electromagnetic problems, where electrically fine structures with details much smaller than a typical wavelength and electrically coarse structures comparable to or larger than a typical wavelength coexist.

Simulations of multiscale electromagnetic problems, such as electromagnetic interference (EMI), electromagnetic compatibility (EMC), and electronic packaging, can be very challenging for conventional numerical methods. In terms of spatial discretization, conventional methods use a single mesh for the whole structure, thus a high discretization density required to capture the geometric characteristics of electrically fine structures will inevitably lead to a large number of wasted unknowns in the electrically coarse parts. This issue will become especially severe for orthogonal grids used by the popular finite-difference time-domain (FDTD) method. In terms of temporal integration, dense meshes in electrically fine domains will make the time step size extremely small for numerical methods with explicit time-stepping schemes. Implicit schemes can surpass stability criterion limited by the Courant-Friedrichs-Levy (CFL) condition. However, due to the large system matrices generated by conventional methods, it is almost impossible to employ implicit schemes to the whole structure for time-stepping.

To address these challenges, we propose an efficient hybrid SETD/FETD method for transient electromagnetic simulations by taking advantages of the strengths of these two methods while avoiding their weaknesses in multiscale problems. More specifically, a multiscale structure is divided into several subdomains based on the electrical size of each part, and a hybrid spectral-element / finite-element scheme is proposed for spatial discretization. The hexahedron-based spectral elements with higher interpolation degrees are efficient in modeling electrically coarse structures, and the tetrahedron-based finite elements with lower interpolation degrees are flexible in discretizing electrically fine structures with complex shapes. A non-spurious finite element method (FEM) as well as a non-spurious spectral element method (SEM) is proposed to make the hybrid SEM/FEM discretization work. For time integration we employ hybrid implicit / explicit (IMEX) time-stepping schemes, where explicit schemes are used for electrically coarse subdomains discretized by coarse spectral element meshes, and implicit schemes are used to overcome the CFL limit for electrically fine subdomains discretized by dense finite element meshes. Numerical examples show that the proposed hybrid SETD/FETD method is free of spurious modes, is flexible in discretizing sophisticated structure, and is more efficient than conventional methods for multiscale electromagnetic simulations.

Item Open Access Advanced SERS Sensing System With Magneto-Controlled Manipulation Of Plasmonic Nanoprobes(2012) Khoury, Christopher GThere is an urgent need to develop practical and effective systems to detect diseases, such as cancer, infectious diseases and cardiovascular diseases.

Nanotechnology is a new, maturing field that employs specialized techniques to detect and diagnose infectious diseases. To this end, there have been a wealth of techniques that have shown promising results, with fluorescence and surface-enhanced Raman scattering being two important optical modalities that are utilized extensively. The progress in this specialized niche is staggering and many research groups in academia, as well as governmental and corporate organizations, are avidly pursuing leads which have demonstrated optimistic results.

Although much fundamental science is still in the pipeline under the guise of both ex-vivo and in-vivo testing, it is ultimately necessary to develop diagnostic devices that are able to impact the greatest number of people possible, in a given population. Such systems make state-of-the-art technology platforms accessible to a large population pool. The development of such technologies provide opportunities for better screening of at-risk patients, more efficient monitoring of disease treatment and tighter surveillance of recurrence. These technologies are also intrinsically low cost, facilitating the large scale screening for disease prevention.

Fluorescence has long been established as the optical transduction method of choice, because of its wealth of available dyes, simple optical system, and long heritage from pathology. The intrinsic limitations of this technique, however, have given rise to a complementary, and more recent, modality: surface-enhanced Raman scattering (SERS). There has been an explosive interest in this technique for the wealth of information it provides without compromising its narrow spectral width.

A number of novel studies and advances are successively presented throughout this study, which culminate to an advanced SERS-based platform in the last chapter.

The finite element method algorithm is critically evaluated against analytical solutions as a potential tool for the numerical modeling of complex, three-dimensional nanostructured geometries. When compared to both the multipole expansion for plane wave excitation, and the Mie-theory with dipole excitation, this algorithm proves to provide more spatially and spectrally accurate results than its alternative, the finite-difference time domain algorithm.

Extensive studies, both experimental and numerical, on the gold nanostar and Nanowave substrate for determining their potential as SERS substrates, constituted the second part of this thesis. The tuning of the gold nanostar geometry and plasmon band to optimize its SERS properties were demonstrated, and significant 3-D modeling was performed on this exotic shape to correlate its geometry to the solution's exhibited plasmon band peak position and large FWHM. The Nanowave substrate was experimentally revived and its periodic array of E-field hotspots, which was until recently only inferred, was finally demonstrated via complex modeling.

Novel gold- and silver- coated magnetic nanoparticles were synthesized after extensive tinkering of the experimental conditions. These plasmonics-active magnetic nanoparticles were small and displayed high stability, were easy to synthesize, exhibited a homogeneous distribution, and were easily functionalizable with Raman dye or thiolated molecules.

Finally, bowtie-shaped cobalt micromagnets were designed, modeled and fabricated to allow the controllable and reproducible concentrating of plasmonics-active magnetic nanoparticles. The external application of an oscillating magnetic field was accompanied by a cycling of the detected SERS signal as the nanoparticles were concentrated and re-dispersed in the laser focal spot. This constituted the first demonstration of magnetic-field modulated SERS; its simplicity of design, fabrication and operation opens doors for its integration into diagnostic devices, such as a digital microfluidic platform, which is another novel concept that is touched upon as the final section of this thesis.

Item Open Access Behavior of different numerical schemes for random genetic drift(BIT Numerical Mathematics, 2019-09-01) Xu, S; Chen, M; Liu, C; Zhang, R; Yue, XIn the problem of random genetic drift, the probability density of one gene is governed by a degenerated convection-dominated diffusion equation. Dirac singularities will always be developed at boundary points as time evolves, which is known as the fixation phenomenon in genetic evolution. Three finite volume methods: FVM1-3, one central difference method: FDM1 and three finite element methods: FEM1-3 are considered. These methods lead to different equilibrium states after a long time. It is shown that only schemes FVM3 and FEM3, which are the same, preserve probability, expectation and positiveness and predict the correct probability of fixation. FVM1-2 wrongly predict the probability of fixation due to their intrinsic viscosity, even though they are unconditionally stable. Contrarily, FDM1 and FEM1-2 introduce different anti-diffusion terms, which make them unstable and fail to preserve positiveness.Item Open Access Computational Modeling of Epidural Cortical Stimulation: Design, Analysis, and Experimental Evaluation(2011) Wongsarnpigoon, AmornEpidural cortical stimulation (ECS) is a developing therapy for many neurological disorders. However, the mechanisms by which ECS has its effects are unknown, and this lack of understanding has limited the development and optimization of this promising therapy. This dissertation investigates the effects of ECS on the neurons in the cortex and how these effects vary with electrode geometry and location as well as the electrical and geometrical properties of the anatomy.

The effects of ECS on cortical neurons were investigated using a three dimensional computational model of the human precentral gyrus and surrounding anatomy. An epidural electrode was placed above the gyrus, and the model was solved using the finite element method. The outputs of the model included distributions of electric potential, the second spatial derivative of potential (activating function), and current density. The distributions of electric potential were coupled to compartmental models of cortical neurons to quantify the effects of ECS on cortical neurons. A sensitivity analysis was performed to assess how thresholds and distributions of activating function were impacted by changes in the geometrical and electrical properties of the model. In vivo experiments of epidural electrical stimulation of cat motor cortex were performed to measure the effects of stimulation parameters and electrode location on thresholds for evoking motor responses.

During ECS, the region of cortex directly underneath the electrode was activated at the lowest thresholds, and neurons deep in the sulcus could not be directly activated without coactivation of neurons located on the crowns or lips of the gyri. The thresholds for excitation of cortical neurons depended on stimulation polarity as well as the orientation and position of the neurons with respect to the electrode. In addition, the patterns and spatial extent of activation were influenced by the geometry of the cortex and surrounding layers, the dimensions of the electrodes, and the positioning of the lead. In vivo thresholds for evoking motor responses were dependent on electrode location and stimulation polarity, and bipolar thresholds were often different from monopolar thresholds through the respective anode and cathode individually. The effects of stimulation polarity and electrode location on thresholds for evoking motor responses paralleled results of the computational model. Experimental evidence of indirect effects of ECS, mediated by synaptic interactions between neural elements, revealed an opportunity for further development of the computational model. The outcome of this dissertation is an improved understanding of the factors that influence the effects of ECS on cortical neurons, and this knowledge will help facilitate the rational implantation and programming of ECS systems.

Item Embargo Efficient Simulations of Electromagnetic Induction Tool in a Deviated Borehole for Resistivity Inversions(2022) Zhong, YangFor the petroleum industry, layered medium subsurface detection plays an important role in discovering reservoirs and drilling wells. In geophysics, resistivity is an essential property for distinguishing formation layers or even small fractures. Well logging with electromagnetic induction tools can measure the subsurface resistivity. This measurement includes two steps: 1) directly measure the low-frequency response signals using the tool and 2) determine the subsurface geometric model and resistivity. The problem is that no formula can directly calculate the resistivity from the measured tool responses. A systematic solution is to combine forward electromagnetic simulations and inversion of the subsurface model. In this dissertation, two categories of inversion are investigated: Determine the proper subsurface model by 1) optimizing the objective function, such as data misfit, and 2) training a surrogate model for the inverse mapping. Many forward simulations are demanded for either estimating the data misfit of new candidate models or collecting data for training. Therefore, efficient electromagnetic simulation is critical for resistivity logging. From complex to simple, three types of simulation are discussed: 1) borehole simulation with real tool configuration, 2) borehole simulation with point sources as the virtual tool, and 3) simplified layered medium simulation with virtual tool. Three optimal methods are implemented, respectively: the domain decomposition method, the finite element boundary integral method, and the analytical method. The tool calibration and the borehole effects are studied in the comparison of these simulations. Ideally, the simplest forward simulation should be used in the inversion, and the additional effects can be extracted as correction terms. The optimization-based inversion of the formation model uses simulations of a virtual tool in the layered medium. The Occam inversion or Monte Carlo Markov chain can minimize the data misfit. Another special simulation for small fractures using the thin dielectric sheet approximate method collects the dataset of fracture models. Fracture parameters such as resistivity, extension, and tilt angle are accurately determined by machine learning methods. The surrogate model also tends to predict fracture properties correctly, even for the complete simulation result.

Item Open Access Finite Element Modeling of Biological Systems(2023) Golshaei, BehzadMechanical properties have a decisive role in the fundamental functions of biological systems, including migration of cells, cell apoptosis, and proliferation of cells and bacteria. This is also true for cancer metastasis and morphogenetic processes during embryonic development. It isn’t easy, however, to study biological systems due to their complex behavior, such as their activity and nonlinear material properties. Note that while the individual mechanical properties of specific biological systems, such as biopolymers, have been well established, the collective behavior of these elements has a different response, as the comparative studies of the mechanical properties of single cancer cells and cancerous tissue demonstrate. Thus, numerous experimental instruments have been developed over the years to investigate biological systems’ mechanical properties, individually or collectively. These experimental techniques can evaluate mechanical properties at multiple scales. Theycan target individual biological entities, like single cells, or assess the collective mechanical properties of more complex biological systems, such as tissues or organoids. The resolution of these studies ranges from single-cell analyses to those concerning embryonic morphogenesis. Simulating biological systems’ individual or collective behavior using a discretized approach (i.e., Molecular Dynamics) or a continuum approach (i.e., Finite Element) is an adjunct to experimental studies. This thesis explores the collective behavior observed in individual cells and embryonic tissue. This exploration was carried out through the development of experimental protocols and the application of continuum mechanics models. In the initial two chapters of this thesis, we delve into the fundamental mechanical concepts essential for understanding the mechanical properties of cells and tissues. We also discuss prior studies that employed shell mechanics to model cellular and embryonic deformations. In the third chapter, we detail our collaborative work with Dr. Samaneh Rezvani focuses on the role of the actin cortex in the deformation of individual suspended spherical cells. For this purpose, we utilized double-trap optical tweezers in conjunction with a viscoelastic pressurized-thick-shell model. Using our simulation approach, we determined the mechanical properties of the actin cortex from the experimental results. The elastic shear modulus of the actin cortex ranged between 4.5 kPa and 7.5 kPa. In modeling the steady deformation of single cells with the shell model, we observed that cell volume remains conserved during deformation. Instead of reducing volume, cells extend the actin cortex to accommodate the increased surface area. We also introduced a multilayer viscoelastic shell model to examine the time-dependent mechanical behaviors of cells, focusing on hysteresis due to dissipative processes. Our model incorporated a fluid core within a viscoelastic shell, offering a more thorough understanding of cell mechanics. Our findings indicate that the damping response in cells is predominantly influenced by the viscosity of the cytosol rather than that of the actin cortex. The fourth chapter describes the modeling of experiments conducted by Dr. Renata Garces on gram-negative E. coli bacteria uniaxially compressed between parallel plates. We used Finite Element Modeling (FEM) to examine the collective mechanical behavior of the peptidoglycan layer (PG) in the bacterial cell wall, modeled as a thin, pressurized rod-shaped shell. Finally, in chapter five, we investigated, in collaboration with Dr. Chonglin Guan, the cells’ collective behavior in epithelial tissue during dorsal closure (DC) in developing Drosophila melanogaster embryos (DME). Utilizing glass microprobes, we deformed various tissue types, specifically amnioserosa (AS) and lateral epidermis (LE), and subsequently recorded their responses to assess the impact of tissue mechanical properties on embryonic development. We simulated a viscoelastic flat shell, replicating the geometry of individual embryos, using the Finite Element Method (FEM) to model tissue deformations. Through this methodology, we quantified the mechanical characteristics of amnioserosa and lateral epidermis, encompassing both their viscosity and elasticity. Our analyses determined the elasticity of AS to be approximately (110 to 180 kPa) and its viscosity to be (0.86 to 1.05 Pa.s). Additionally, we executed step-function experiments to ascertain tissue mechanical properties and evaluate tissue relaxation time. Our findings are in line with our previous results obtained from hysteresis studies.

Item Open Access Influence of Material Properties and Fracture Properties on the Crack Nucleation and Growth(2021) Zeng, BoIn this thesis, we studied the influence of spatial variations in the fracture property and the elastic property on the resulting crack patterns during soil desiccation. Young's modulus is selected as the representative elastic property and the fracture toughness is selected as that for the fracture property. Their well-defined spatially fluctuated random fields are the input of the phase-field fracture simulation, and the resulting damage field is the output. Various postprocessing of the damage field were carried out to analyze the resulting fields. After comparing the morphology of the cracks and fragment size distributions, a preliminary guess was that the two inputs have very close influence on the output. Then the Pearson correlation coefficient, as a first try of sensitivity analysis, also gave an indistinguishable correlation number between the two. A more rigorous approach with highly isolated sensitivity quantity was needed, which brought us to the Sobol' indice based on polynomial chaos expansion, a global sensitivity analysis measure which accounts for the variation of output into the variation of each input and any combination of input.

Item Open Access Multi-Scale Modeling for Analysis and Design of Transcranial Electric and Magnetic Brain Stimulation(2021) Aberra, Aman SenayTranscranial electric stimulation (tES) and magnetic stimulation (TMS) can noninvasively modulate brain activity in humans, offering broad research and therapeutic applications. However, improving the efficacy and selectivity of these techniques is challenging without a mechanistic understanding of how the stimulation parameters determine the neural response and how these parameters can be manipulated to activate specific neural circuits. This dissertation presents multiscale computational models that predict the neural response to TMS and tES at the single-cell and population levels for analysis and rational design of transcranial brain stimulation.We adapted biophysically-realistic models of cortical neurons from the Blue Brain network to the properties of mature rat and human neurons and characterized the direct response to extracellular stimulation with both subthreshold and suprathreshold electric field (E-field) stimulation modalities. These models included 3D reconstructed axonal and dendritic arbors as well as multiple excitatory and inhibitory cell types with validated electrophysiological behavior. Axon terminals were the lowest threshold elements for stimulation, and their dependence on threshold and polarization was determined by cell-type specific morphological features, such as myelination, diameter, and branching. However, we found for TMS pulse waveforms specifically, activation thresholds were higher than expected from in vivo applications. We improved the fidelity of the axon models further using a feature-based optimization algorithm, but these modifications did not produce models with significantly lower thresholds, suggesting other factors may allow for suprathreshold activation at the E-field intensities induced experimentally. The neuron models were then embedded in anatomically-realistic volume conductor head models of the E-field in humans derived from magnetic resonance imaging (MRI) data to simulate the direct neural response to TMS and tDCS. The models reproduced relative trends in motor thresholds as well as the experimentally-measured strength–duration time constant. TMS activated with lowest intensity intracortical axon terminals in the superficial gyral crown and lip region, proportional to the E-field magnitude. Thresholds were lowest for the L5 pyramidal cells (PCs), with activation of the L2/3 PCs and large basket cells at most intensities. Reversing the pulse direction revealed waveform-dependent spatial shifts in the activated neural population that may explain experimentally observed differences in the latencies and thresholds of muscle responses to TMS of motor cortex. We also quantified the subthreshold polarization generated by conventional tDCS with large rectangular pad electrodes and 4×1 high definition (HD) tDCS electrodes targeting the motor hand knob. Axonal and dendritic terminal polarization was higher than somatic polarization in all cell types, and polarization trends between cell types varied by subcellular compartment. While the HD tDCS montage produced a significantly more focal E-field within the brain, both montages generated broad regions of depolarization and hyperpolarization beneath the electrodes. These simulations demonstrated the importance of coupling the E-field to neuron models incorporating non-linear membrane dynamics and realistic morphologies for predicting the neural response to TMS and tES. Indeed, extrapolating the neural response (polarization or threshold) from the uniform E-field or macroscopic E-field components often led to erroneous predictions. Due to the high computational cost of these biophysically-realistic models, we also developed rapid estimators of the neural response using a 3D convolutional neural network. This approach allowed for reproducing the threshold distributions of the realistic model neurons with several orders of magnitude shorter run times than using the E-field distribution alone. In sum, this work provides both computational tools and mechanistic insights to improve the use and development of transcranial magnetic and electrical stimulation technologies.

Item Open Access Stable Embedded Grid Techniques in Computational Mechanics(2010) Sanders, JessicaEngineering mechanics problems involving interfaces, whether physical or introduced by numerical methodologies, are commonplace. Just a few examples include fracture and fault mechanics, classical contact-impact, phase boundary propagation, and fluid-structure interaction. This dissertation focuses on issues of numerical stability and accuracy that must be addressed when such interfaces are included in a realistic simulation of a physical system.

We begin by presenting a novel numerical method of fluid/structure interaction that may be applied to the problem of movable devices and ocean waves. The work is done with finite differences, large motion Lagrangian mechanics, and an eye towards creating a model in which complex rigid body dynamics can be incorporated.

We then review the many advantages of embedded mesh techniques for interface representation, and investigate a completely finite element based strategy for embedding domains. The work is seen as a precursor to robust multi-physics simulation capabilities. Special attention must be given to these techniques in terms of stable and convergent representation of surface fluxes. Mesh locking and over-constraint are particularly addressed. We show that traditional methods for enforcing continuity at embedded interfaces cannot adequately guarantee flux stability, and show a less traditional method, known as Nitsche's method, to be a stable alternative. We address the open problem of applying Nitsche's method to non-linear analysis by drawing parallels between the embedded mesh and discontinuous Galerkin (DG) methods, and propose a DG style approach to Nitsche's method. We conclude with stable interfacial fluxes for a continuity constraint for a case of embedded finite element meshes in large deformation elasticity. The general conclusion is drawn that stability must be addressed in the choice of interface treatment in computational mechanics.

Item Open Access The Effect of Wing Damage on Aeroelastic Behavior(2009) Conyers, Howard J.Theoretical and experimental studies are conducted in the field of aeroelasticity. Specifically, two rectangular and one cropped delta wings with a hole are analyzed in this dissertation for their aeroelastic behavior.

The plate-like wings are modeled using the finite element method for the structural theory. Each wing is assumed to behave as a linearly elastic and isotropic, thin plate. These assumptions are those of small-deflection theory of bending which states that the plane sections initially normal to the midsurface remain plane and normal to that surface after bending. The wings are modeled in low speed flows according to potential flow theory. The potential flow is governed by the aerodynamic potential equation, a linear partial differential equation. The aerodynamic potential equation is solved using a distribution of doublets that relates pressure to downwash in the doublet lattice method. A hole in a wing-like structure is independently investigated theoretically and experimentally for its structural and aerodynamic behavior.

The aeroelastic model couples the structural and aerodynamic models using Lagrange's equations. The flutter boundary is predicted using the V-g method. Linear theoretical models are capable of predicting the critical flutter velocity and frequency as verified by wind tunnel tests. Along with flutter prediction, a brief survey on gust response and the addition of stores(missile or fuel tanks) are examined.

Item Open Access The Second Generation Shifted Boundary Method with Applications to Porous Media Flow and Solid Mechanics(2020) Atallah, NabilComplex geometries has been a challenge to numerical algorithms. For classical body-fitted computational techniques, the challenge manifests itself in the time consuming and labor intensive grid generation phase. While for standard embedded/immersed methods, the challenge is mainly in the complicated and computationally intensive geometric construction of the partial elements cut by the embedded boundary.

Recently, the shifted boundary method (SBM) was proposed by Main and Scovazzi within the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids the geometric construction of the cut elements and maintains accuracy by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions.

The first part of this thesis is devoted to the development and numerical analysis of the enhanced variational SBM formulations for the Poisson and Stokes problems over their original counterparts. First, we show that these second-generation SBM implementations can be proved asymptotically stable and convergent without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive. Second, we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at Dirichlet boundaries, therefore avoiding the calibration of an additional stabilization parameter. Finally, we prove enhanced L2-estimates without the cumbersome assumption - of earlier proofs - that the surrogate domain is convex.

As for the second part of the thesis, we adopt the second generation formulations as a reference point and propose a new SBM framework for the flow in porous media (Darcy flow) equations. In particular, we develop equal-order discontinuous Galerkin (DG) in addition to continuous Galerkin (CG) discretizations to accurately capture the velocity and pressure fields under highly anisotropic and/or heterogeneous porous media. We corroborate our CG and DG schemes with a full analysis of stability and convergence in addition to extensive tests in two and three dimensions. The value of this approach is clearly visible from the 3D simulation of water flow around tree roots which was otherwise not possible with a body-fitted approach.

In third and final part of this thesis, we develop a SBM framework for the solid mechanics equations; in particular, the equations of linear isotropic elastostatics. The main challenge was with handling traction boundary conditions for a displacement-based, irreducible formulation of the SBM in combination with piecewise linear finite element spaces. We circumvented this problem by transforming the displacement-based equation into a mixed strain-displacement one; resulting in a system of equations akin to the Darcy flow one. The net result is a more accurate approximation of stresses and strains in exchange for an increase in the computational and storage costs. If this tradeoff is deemed unacceptable, the mixed formulation is restricted to a layer of elements (of unit depth) in proximity of the surrogate (approximate) boundary, while applying a standard primal formulation everywhere else. The net result is an enhanced formulation that maintains the bulk cost of the base primal formulation, but allows for an accurate imposition of boundary conditions. A full analysis of stability and convergence of the method is presented and complemented with an extensive set of computational experiments in two and three dimensions, for progressively more complex geometries.

Item Open Access The Shifted Interface/Boundary Method for Embedded Domain Computations(2021) Li, KanganNumerical computations involving complex geometries have a wide variety of applications in both science and engineering, including the simulation of fractures, melting and solidification, multiphase flows, biofilm growth, etc. Classical finite element methods rely on computational grids that are adapted (fitted) to the geometry, but this approach creates fundamental computational challenges, especially when considering evolving interfaces/boundaries. Embedded methods facilitate the treatment of complex geometries by avoiding fitted grids in favor of immersing the geometry on pre-existing grids.

The first part of this thesis work introduces a new embedded finite element interface method, the shifted interface method (SIM), to simulate partial differential equations over domains with internal interfaces. Our approach belongs to the family of surrogate/approximate interface methods and relies on the idea of shifting the location and value of interface jump conditions, by way of Taylor expansions. This choice has the goal of preserving optimal convergence rates while avoiding small cut cells and related problematic issues, typical of traditional embedded methods. In this part, SIM is applied to internal interface computations in the context of the mixed Poisson problem, also known as the Darcy flow problem and is extended to linear isotropic elasticity interface problems. An extensive set of numerical tests is provided to demonstrate the accuracy, robustness and efficiency of the proposed approach.

In the second part of the thesis, we propose a new framework for linear fracture mechanics, based on the idea of an approximate fracture geometry representation combined with approximate interface conditions. The approach evolves from SIM, and introduces the concept of an approximate fracture surface composed of the full edges/faces of an underlying grid that are geometrically close to the true fracture geometry. Similar to SIM, the interface conditions on the surrogate fracture are modified with Taylor expansions to achieve a prescribed level of accuracy. This shifted fracture method (SFM) does not require cut cell computations or complex data structures, since the behavior of the true fracture is mimicked with standard integrals on the approximate fracture. Furthermore, the energetics of the true fracture are represented within the prescribed level of accuracy and independently of the grid topology. We present a general computational framework and then apply it in the specific context of cohesive zone models, with an extensive set of numerical experiments in two and three dimensions.

In the third and final part, we develop a shifted boundary method (SBM), originated from Main and Scovazzi (2018), for the thermoelasticity equations. SBM requires to shift the location and value of both Dirichlet and Neumann boundary conditions to surrogate boundaries with Taylor expansions. In such a way, an opti- mal convergence rate can be preserved for both temperature and displacement. An extensive of numerical examples in both two and three dimensions are presented in this part to demonstrate the performance of SBM.