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dc.contributor.authorGuo, Ruchien_US
dc.date.accessioned2019-06-20T08:01:23Z
dc.date.available2019-06-20T08:01:23Z
dc.date.issued2019-06-19
dc.identifier.othervt_gsexam:19765en_US
dc.identifier.urihttp://hdl.handle.net/10919/90374
dc.description.abstractThis dissertation consists of three studies of immersed finite element (IFE) methods for inter- face problems related to partial differential equations (PDEs) with discontinuous coefficients. These three topics together form a continuation of the research in IFE method including the extension to elasticity systems, new breakthroughs to higher degree IFE methods, and its application to inverse problems. First, we extend the current construction and analysis approach of IFE methods in the literature for scalar elliptic equations to elasticity systems in the vector format. In particular, we construct a group of low-degree IFE functions formed by linear, bilinear, and rotated Q1 polynomials to weakly satisfy the jump conditions of elasticity interface problems. Then we analyze the trace inequalities of these IFE functions and the approximation capabilities of the resulted IFE spaces. Based on these preparations, we develop a partially penalized IFE (PPIFE) scheme and prove its optimal convergence rates. Secondly, we discuss the limitations of the current approaches of IFE methods when we try to extend them to higher degree IFE methods. Then we develop a new framework to construct and analyze arbitrary p-th degree IFE methods. In this framework, each IFE function is the extension of a p-th degree polynomial from one subelement to the whole interface element by solving a local Cauchy problem on interface elements in which the jump conditions across the interface are employed as the boundary conditions. All the components in the analysis, including existence of IFE functions, the optimal approximation capabilities and the trace inequalities, are all reduced to key properties of the related discrete extension operator. We employ these results to show the optimal convergence of a discontinuous Galerkin IFE (DGIFE) method. In the last part, we apply the linear IFE methods in the literature together with the shape optimization technique to solve a group of interface inverse problems. In this algorithm, both the governing PDEs and the objective functional for interface inverse problems are discretized optimally by the IFE method regardless of the location of the interface in a chosen mesh. We derive the formulas for the gradients of the objective function in the optimization problem which can be implemented efficiently in the IFE framework through a discrete adjoint method. We demonstrate the properties of the proposed algorithm by applying it to three representative applications.en_US
dc.format.mediumETDen_US
dc.publisherVirginia Techen_US
dc.rightsThis item is protected by copyright and/or related rights. Some uses of this item may be deemed fair and permitted by law even without permission from the rights holder(s), or the rights holder(s) may have licensed the work for use under certain conditions. For other uses you need to obtain permission from the rights holder(s).en_US
dc.subjectElliptic Interface Problemsen_US
dc.subjectElasticity Interface Problemsen_US
dc.subjectUnfitted Meshesen_US
dc.subjectImmersed Finite Elementen_US
dc.subjectHigh-order Discontinuous Galerkin methodsen_US
dc.subjectError Analysisen_US
dc.subjectInterface Inverse Problemsen_US
dc.subjectShape Optimizationen_US
dc.titleDesign, Analysis, and Application of Immersed Finite Element Methodsen_US
dc.typeDissertationen_US
dc.contributor.departmentMathematicsen_US
dc.description.degreeDoctor of Philosophyen_US
thesis.degree.nameDoctor of Philosophyen_US
thesis.degree.leveldoctoralen_US
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen_US
thesis.degree.disciplineMathematicsen_US
dc.contributor.committeechairLin, Taoen_US
dc.contributor.committeememberAdjerid, Slimaneen_US
dc.contributor.committeememberYue, Pengtaoen_US
dc.contributor.committeememberWarburton, Timothyen_US
dc.description.abstractgeneralInterface problems arise from many science and engineering applications modeling the transmission of some physical quantities between multiple materials. Mathematically, these multiple materials in general are modeled by partial differential equations (PDEs) with discontinuous parameters, which poses challenges to developing efficient and reliable numerical methods and the related theoretical error analysis. The main contributions of this dissertation is on the development of a special finite element method, the so called immersed finite element (IFE) method, to solve the interface problems on a mesh independent of the interface geometry which can be advantageous especially when the interface is moving. Specifically, this dissertation consists of three projects of IFE methods: elasticity interface problems, higher-order IFE methods and interface inverse problems, including their design, analysis, and application.en


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