Isogeometric analysis is an innovative numerical paradigm with the potential to bridge the gap between Computer-Aided Design and Computer-Aided Engineering. However, constructing analysis-suitable parameterizations from a given boundary representation remains a critical challenge in the isogeometric design-through-analysis pipeline, particularly for computational domains with complex geometries, such as high-genus cases. To tackle this issue, we propose a multi-patch parameterization method for computational domains grounded in the singular structure of cross-fields. Initially, the vector field functions over the computational domain are solved using the boundary element method. The cross-field is then obtained through the one-to-one mapping between the vector field and the cross-field. Subsequently, we acquire the position information and topological connection relations of singularities and streamlines by analyzing the singular structure of the cross-field. Moreover, we introduce a simple and effective method for computing streamlines. We propose a novel segmentation strategy to divide the computational domain into several quadrilateral NURBS sub-patches. Once the multi-patch structure is established, we develop two methods to construct analysis-suitable multi-patch parameterizations. The first method is a direct generalization of the barrier function-based approach, while the second method yields smoother parameterizations by incorporating the interface control points of sub-patches into the optimization model. Numerical experiments demonstrate the effectiveness and robustness of the proposed method.@en

Structure-conserving numerical methods that aim at preserving certain structures of the PDEs at the discrete level have been an interesting research topic for many decades. The mimetic spectral element method, a recently developed arbitrary order structure-preserving method on orthogonal or curvilinear meshes, has also been drawing increasingly amount of attention. This dissertation is devoted to promoting the application and development of the mimetic spectral element method. In this dissertation, we first give a comprehensive introduction on the mimetic spectral element method with applications to the Poisson problem, which is followed by a new development of the mimetic spectral element method for the Navier-Stokes equations. This new development is on a conservative dual-field discretization that conserves mass, kinetic energy and helicity for the 3D incompressible Navier-Stokes equations in the absence of dissipative terms. And when there are dissipative terms, the method correctly predicts the decay rates of the kinetic energy and helicity. It is a dual-field method in the sense that two evolution equations are employed and weak solutions are sought for each physical variable in two different finite dimensional function spaces. This novel method and the promising results reveal its potential in multiple research fields like turbulence modeling, sub-grid methods and large eddy simulation. Despite the mimetic spectral element method possesses preferable properties due to its feature of structure-preserving, its demand of high computational power is a major limitation. To address this drawback, two techniques, hybridization and dual basis functions, are employed for the mimetic spectral element method, which leads to an extension that decreases the computational cost not only by reducing the size and lowering the condition number of the global linear system, but also by improving the feasibility for parallel computing. A special component, the Complement, is embedded in this thesis. It aims to provide a more friendly introduction for the readers, especially those who are new to this specific area of numerical methods. In these web-based additions, there are instructors and well-documented scripts which allow readers to learn in an interactive way, thus to get some hands-on experience and eventually to obtain a deeper understanding of the method. This component can help the readers to more quickly and efficiently implement their own new ideas, which will in return contribute to the development of this method. Overall, we conclude that this dissertation fulfilled the goal to promote the application and development of the mimetic spectral element method.@en

We introduce a domain decomposition structure-preserving method based on a hybrid mimetic spectral element method for three-dimensional linear elasticity problems in curvilinear conforming structured meshes. The method is an equilibrium method which satisfies pointwise equilibrium of forces. The domain decomposition is established through hybridization which first allows for an inter-element normal stress discontinuity and then enforces the normal stress continuity using a Lagrange multiplier which turns out to be the displacement in the trace space. Dual basis functions are employed to simplify the discretization and to obtain a higher sparsity. Numerical tests supporting the method are presented.

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In ℝⁿ, let Λ^k(Ω) represent the space of smooth differential k-forms in Ω. The de Rham complex consists of a sequence of spaces, Λ^k(Ω), k = 0, 1…, n, connected by the exterior derivative, d: Λ^k(Ω) → Λ^k+1(Ω). Appropriately chosen B-spline spaces together with their associated dual B-spline spaces form a discrete double de Rham complex. In practical applications, this discrete double de Rham complex leads to very sparse systems. In this paper, this construction will be explained and illustrated by means of a non-trivial three-dimensional example.

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In this paper, we present a hybrid mimetic method which solves the mixed formulation of the Poisson problem on curvilinear quadrilateral meshes. The method is hybrid in the sense that the domain is decomposed into multiple disjoint elements and the interelement continuity is enforced using a Lagrange multiplier. The method is mimetic in the sense that the discrete divergence operator is exact. By using the mimetic basis functions and their dual representations, various metric-free discrete terms are obtained. The discrete system can be efficiently solved by first solving a reduced system for the Lagrange multiplier. Numerical experiments which validate the method are presented.

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Given a sequence of finite element spaces which form a de Rham sequence, we will construct dual representations of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The dual representations also need to satisfy the de Rham sequence on the domain boundary. The matrix which converts primal representations to dual representations – the Hodge matrix – is the mass or Gram matrix. It will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients (degrees of freedom) of both representations. This leads to very sparse system matrices, even for high order methods. The differential operators for dual representations will be defined. Vector operations, grad, curl and div, for primal and dual representations are both topological and do not depend on the metric, i.e. the size and shape of the mesh or the order of the numerical method. Derivatives are evaluated by applying sparse incidence and inclusion matrices to the expansion coefficients of the representations. As illustration of the use of dual representations, the method will be applied to (i) a mixed formulation for the Poisson problem in 3D, (ii) it will be shown that this approach allows one to preserve the equivalence between Dirichlet and Neumann problems in the finite dimensional setting and, (iii) the method will be applied to the approximation of grad–div eigenvalue problem on affine and non-affine meshes.

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In this paper we will consider two curl-curl equations in two dimensions. One curl-curl problem for a scalar quantity F and one problem for a vector field E. For Dirichlet boundary conditions n× E= Ê⊣ on E and Neumann boundary conditions n×curlF=Ê⊣, we expect the solutions to satisfy E = curl F. When we use algebraic dual polynomial representations, these identities continue to hold at the discrete level. Equivalence will be proved and illustrated with a computational example.@en

In this paper, we will use algebraic dual polynomials to set up a discrete Steklov-Poincaré operator for the mixed formulation of the Poisson problem. The method will be applied in curvilinear coordinates and to a test problem which contains a singularity. Exponential convergence of the trace variable in H 1 / 2 {H^{1/2}} -norm will be shown.@en

This chapter addresses the topological structure of steady, anisotropic, inhomogeneous diffusion problems. Differential operators are represented by sparse incidence matrices, while weighted mass matrices play the role of metric-dependent Hodge matrices. The resulting mixed formulation is point-wise divergence-free if the right hand side function f = 0. The method is inf-sup stable; no stabilization is required and the method displays optimal convergence on orthogonal and deformed grids.

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