Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft

Computational Gradient Elasticity and Gradient Plasticity with Adaptive Splines

Classical continuum mechanics theories are largely insufficient in capturing size effects observed in many engineering materials: metals, composites, rocks etc. This is attributed to the absence of a length scale that accounts for microstructural effects inherent in these materials. Enriching the classical theories with an internal length scale solves this problem. One way of doing this, in a theoretically sound manner, is introducing higher order gradient terms in the constitutive relations. In elasticity, introducing a length scale removes the singularity observed at crack tips using the classical theory. In plasticity, it eliminates the spurious mesh sensitivity observed in softening and localisation problems by defining the width of the localisation zone thereby maintaining a well-posed boundary value problem. However, this comes at the cost of more demanding solution techniques. Higher-order continuity is usually required for solving gradient-enhanced continuum theories, a requirement difficult to meet using traditional finite elements. Hermitian finite elements, mixed methods and meshless methods have been developed to meet this requirement, however these methods have their drawbacks in terms of efficiency, robustness or implementational convenience. Isogeometric analysis, which exploits spline-based shape functions, naturally incorporates higher-order continuity, in addition to capturing the exact geometry and expediting the design-through-analysis process. Despite its potentials, it is yet to be fully explored for gradient-enhanced continua. Hence, this thesis develops an isogeometric analysis framework for gradient elasticity and gradient plasticity. The linearity of the gradient elasticity formulation has enabled an operator-split approach so that instead of solving the fourth-order partial differential equation monolithically, a set of two partial differential equations is solved in a staggered manner. A detailed convergence analysis is carried out for the original system and the split set using NURBS and T-splines. Suboptimal convergence rates in the monolithic approach and the limitations of the staggered approach are substantiated. Another advantage of the spline-based approach adopted in this work is the ease with which different orders of interpolation can be achieved. This is useful for consistency, and relevant in gradient plasticity where the local (explicit formulation) or nonlocal (implicit formulation) effective plastic strain needs to be discretised in addition to the displacements. Using different orders of interpolation, both formulations are explored in the second-order and a fourth-order implicit gradient formulation is proposed. Results, corroborated by dispersion analysis, show that all considered models give good regularisation with mesh-independent results. Compared with finite element approaches that use Hermitian shape functions for the plastic multiplier or mixed finite element approaches, isogeometric analysis has the distinct advantage that no interpolation of derivatives is required. In localisation problems, numerical accuracy requires the finite element size employed in simulations to be smaller than the internal length scale. Fine meshes are also needed close to regions of geometrical singularities or high gradients. Maintaining a fine mesh globally can incur high computational cost especially when considering large structures. To achieve this efficiently, selective refinement of the mesh is therefore required. In this context, splines need to be adapted to make them analysis-suitable. Thus, an adaptive isogeometric analysis framework is also developed for gradient elasticity and gradient plasticity. The proposed scheme does not require the mesh size to be smaller than the length scale, even during analysis, until a localisation band develops upon which adaptive refinement is performed. Refinement is based on a multi-level mesh with truncated hierarchical basis functions interacting through an inter-level subdivision operator. Through Bezier extraction, truncation of the bases is simplified by way of matrix multiplication, and an element-wise standard finite element data structure is maintained. In sum, a robust computational framework for engineering analysis is established, combining the flexibility, exact geometry representation, and expedited design-through analysis of isogeometric analysis, size-effect capabilities and mesh-objective results of gradient-enhanced continua, the standard convenient data structure of finite element analysis and the improved efficiency of adaptive hierarchical refinement

Non-Markovian dynamics in open quantum systems

The dynamical behavior of open quantum systems plays a key role in many applications of quantum mechanics, examples ranging from fundamental problems, such as the environment-induced decay of quantum coherence and relaxation in many-body systems, to applications in condensed matter theory, quantum transport, quantum chemistry and quantum information. In close analogy to a classical Markov process, the interaction of an open quantum system with a noisy environment is often modelled by a dynamical semigroup with a generator in Lindblad form, which describes a memoryless dynamics leading to an irreversible loss of characteristic quantum features. However, in many applications open systems exhibit pronounced memory effects and a revival of genuine quantum properties such as quantum coherence and correlations. Here, recent results on the rich non-Markovian quantum dynamics of open systems are discussed, paying particular attention to the rigorous mathematical definition, to the physical interpretation and classification, as well as to the quantification of memory effects. The general theory is illustrated by a series of examples. The analysis reveals that memory effects of the open system dynamics reflect characteristic features of the environment which opens a new perspective for applications, namely to exploit a small open system as a quantum probe signifying nontrivial features of the environment it is interacting with. This article further explores the various physical sources of non-Markovian quantum dynamics, such as structured spectral densities, nonlocal correlations between environmental degrees of freedom and correlations in the initial system-environment state, in addition to developing schemes for their local detection. Recent experiments on the detection, quantification and control of non-Markovian quantum dynamics are also discussed.Comment: 26 pages, 10 figure

Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase-field fracture by nonlocal operator method

The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase-field fracture method. The nonlocal governing equations are expressed as an integral form on support and dual-support. The first example shows that the nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding nonlocal forms. In addition, a criterion based on the instability of the nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate nonlocal elasticity and the nonlocal thin plate. © 2021, The Author(s)

Structure tensor total variation

This is the final version of the article. Available from Society for Industrial and Applied Mathematics via the DOI in this record.We introduce a novel generic energy functional that we employ to solve inverse imaging problems within a variational framework. The proposed regularization family, termed as structure tensor total variation (STV), penalizes the eigenvalues of the structure tensor and is suitable for both grayscale and vector-valued images. It generalizes several existing variational penalties, including the total variation seminorm and vectorial extensions of it. Meanwhile, thanks to the structure tensor’s ability to capture first-order information around a local neighborhood, the STV functionals can provide more robust measures of image variation. Further, we prove that the STV regularizers are convex while they also satisfy several invariance properties w.r.t. image transformations. These properties qualify them as ideal candidates for imaging applications. In addition, for the discrete version of the STV functionals we derive an equivalent definition that is based on the patch-based Jacobian operator, a novel linear operator which extends the Jacobian matrix. This alternative definition allow us to derive a dual problem formulation. The duality of the problem paves the way for employing robust tools from convex optimization and enables us to design an efficient and parallelizable optimization algorithm. Finally, we present extensive experiments on various inverse imaging problems, where we compare our regularizers with other competing regularization approaches. Our results are shown to be systematically superior, both quantitatively and visually

A configurational force for adaptive re-meshing of gradient-enhanced poromechanics problems with history-dependent variables

We introduce a mesh-adaption framework that employs a multi-physical configurational force and Lie algebra to capture multiphysical responses of fluid-infiltrating geological materials while maintaining the efficiency of the computational models. To resolve sharp gradients of both displacement and pore pressure, we introduce an energy-estimate-free re-meshing criterion by extending the configurational force theory to consider the energy dissipation due to the fluid diffusion and the gradient-dependent plastic flow. To establish new equilibria after remeshing, the local tensorial history-dependent variables at the integration points are first decomposed into spectral forms. Then, the principal values and directions are projected onto smooth fields interpolated by the basis function of the finite element space via the Lie-algebra mapping. Our numerical results indicate that this Lie algebra operator in general leads to a new trial state closer to the equilibrium than the ones obtained from the tensor component mapping approach. A new configurational force for dissipative fluid-infiltrating porous materials that exhibit gradient-dependent plastic flow is introduced such that the remeshing may accommodate the need to resolve the sharp pressure gradient as well as the strain localization. The predicted responses are found to be not influenced by the mesh size due to the micromorphic regularization, while the adaptive meshing enables us to capture the width of deformation bands without the necessity of employing fine mesh everywhere in the domain

Nonlocal smoothing and adaptive morphology for scalar- and matrix-valued images

In this work we deal with two classic degradation processes in image analysis, namely noise contamination and incomplete data. Standard greyscale and colour photographs as well as matrix-valued images, e.g. diffusion-tensor magnetic resonance imaging, may be corrupted by Gaussian or impulse noise, and may suffer from missing data. In this thesis we develop novel reconstruction approaches to image smoothing and image completion that are applicable to both scalar- and matrix-valued images. For the image smoothing problem, we propose discrete variational methods consisting of nonlocal data and smoothness constraints that penalise general dissimilarity measures. We obtain edge-preserving filters by the joint use of such measures rich in texture content together with robust non-convex penalisers. For the image completion problem, we introduce adaptive, anisotropic morphological partial differential equations modelling the dilation and erosion processes. They adjust themselves to the local geometry to adaptively fill in missing data, complete broken directional structures and even enhance flow-like patterns in an anisotropic manner. The excellent reconstruction capabilities of the proposed techniques are tested on various synthetic and real-world data sets.In dieser Arbeit beschäftigen wir uns mit zwei klassischen Störungsquellen in der Bildanalyse, nämlich mit Rauschen und unvollständigen Daten. Klassische Grauwert- und Farb-Fotografien wie auch matrixwertige Bilder, zum Beispiel Diffusionstensor-Magnetresonanz-Aufnahmen, können durch Gauß- oder Impulsrauschen gestört werden, oder können durch fehlende Daten gestört sein. In dieser Arbeit entwickeln wir neue Rekonstruktionsverfahren zum zur Bildglättung und zur Bildvervollständigung, die sowohl auf skalar- als auch auf matrixwertige Bilddaten anwendbar sind. Zur Lösung des Bildglättungsproblems schlagen wir diskrete Variationsverfahren vor, die aus nichtlokalen Daten- und Glattheitstermen bestehen und allgemeine auf Bildausschnitten definierte Unähnlichkeitsmaße bestrafen. Kantenerhaltende Filter werden durch die gemeinsame Verwendung solcher Maße in stark texturierten Regionen zusammen mit robusten nichtkonvexen Straffunktionen möglich. Für das Problem der Datenvervollständigung führen wir adaptive anisotrope morphologische partielle Differentialgleichungen ein, die Dilatations- und Erosionsprozesse modellieren. Diese passen sich der lokalen Geometrie an, um adaptiv fehlende Daten aufzufüllen, unterbrochene gerichtet Strukturen zu schließen und sogar flussartige Strukturen anisotrop zu verstärken. Die ausgezeichneten Rekonstruktionseigenschaften der vorgestellten Techniken werden anhand verschiedener synthetischer und realer Datensätze demonstriert