Summary
In this article, we present an extension of the formulation recently developed by the authors to the structural dynamics setting. Inspired by a structure‐preserving family of variational ...integrators, our new formulation relies on a discrete balance equation that establishes the dynamic equilibrium. From this point of departure, we first derive an “exact” discrete‐continuous nonlinear optimization problem that works directly with data sets. We then develop this formulation further into an “approximate” nonlinear optimization problem that relies on a general constitutive model. This underlying model can be identified from a data set in an offline phase. To showcase the advantages of our framework, we specialize our methodology to the case of a geometrically exact beam formulation that makes use of all elements of our approach. We investigate three numerical examples of increasing difficulty that demonstrate the excellent computational behavior of the proposed framework and motivate future research in this direction.
Data-Driven Computational Mechanics is a novel computing paradigm that enables the transition from standard data-starved approaches to modern data-rich approaches. At this early stage of development, ...one can distinguish two mainstream directions. The first one, which can be classified as a direct approach, relies on a discrete-continuous optimization problem and seeks to assign to each material point a point in the phase space that satisfies compatibility and equilibrium, while being closest to the data set provided. The second one, which can be classified as an inverse approach, seeks to reconstruct a constitutive manifold from data sets by manifold learning techniques, relying on a well-defined functional structure of the underlying constitutive law. In this work, we propose a hybrid approach that combines the strengths of the two existing directions and mitigates some of their weaknesses. This is achieved by the formulation of an approximate nonlinear optimization problem, which can be robustly solved, is computationally efficient, and does not rely on any special functional structure of the reconstructed constitutive manifold. Additional benefits include the natural incorporation of kinematic constraints and the possibility to operate with implicitly defined stress–strain relations. We discuss important mathematical aspects of our approach for a data-driven truss element and investigate its key numerical behavior for a data-driven beam element that makes use of all components of our methodology.
•Approximate nonlinear optimization problem for Data-Driven Structural Analysis in general elasticity.•Robust and computationally efficient, not relying on any special functional structure of the constitutive manifold.•Natural incorporation of kinematic constraints.•Possibility to operate with implicitly defined stress–strain relations.•Application of the methodology to the geometrically exact beam.
We describe a local iterative corrector scheme that significantly improves the accuracy of the multiscale finite element method (MsFEM). Our technique is based on the definition of a local corrector ...problem for each multiscale basis function that is driven by the residual of the previous multiscale solution. Each corrector problem results in a local corrector solution that improves the accuracy of the corresponding multiscale basis function at element interfaces. We cast the strategy of residual-driven correction in an iterative scheme that is straightforward to implement and, due to the locality of corrector problems, well-suited for parallel computing. We show that the iterative scheme converges to the best possible fine-mesh solution. Finally, we illustrate the effectiveness of our approach with multiscale benchmarks characterized by missing scale separation, including the microCT-based stress analysis of a vertebra with trabecular microstructure.
This paper presents an analytical solution methodology for the complete stress and displacement fields of a laterally confined granular column loaded from the top end. The granular column is ...idealized as a homogeneous isotropic elastic medium with Coulomb’s friction at the lateral boundary. The solution methodology consists of an analytical procedure that incorporates a potential approach with trigonometric series and Bessel functions, finite Fourier transforms and the superposition method, and an iterative algorithm to satisfy the Coulomb’s friction condition at the lateral boundary. Stress and displacement fields are computed for a specific example and found completely consistent with corresponding finite element results. Key characteristics, computational errors, the convergence behavior, and restrictions of the present approach are discussed. The methodology developed herein can be beneficially applied in the validation process of numerical simulation techniques in granular mechanics such as finite or discrete element methods.
Celotno besedilo
Dostopno za:
DOBA, FGGLJ, IZUM, KILJ, NUK, ODKLJ, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
Following a series of recent innovations, isogeometric shell analysis based on trimmed CAD surfaces is currently being developed into an accurate, efficient and mature design-through-analysis ...methodology. This work contributes to this emerging technology with respect to the following aspects. On the analysis side, we present a robust variationally consistent Nitsche-type formulation for thin shells at large deformations that weakly enforces coupling constraints at trimming curves. On the geometry side, we present a set of algorithms that enable automatic interaction of trimmed shell analysis with CAD data structures based on the STEP exchange format. We integrate these methodologies in a comprehensive framework for isogeometric trimmed shell analysis. We demonstrate that our framework is able to seamlessly perform large-deformation stress analysis of an industry-scale 76-patch surface model of a Dodge RAM hood, while delivering comparable accuracy with respect to Simulia’s commercial software package Abaqus.
•We present a variationally consistent formulation for coupling isogeometric shells at trimming curves.•We also present algorithms that enable automatic interaction with CAD data structures based on the STEP exchange format.•We integrate these methodologies in a comprehensive framework for isogeometric trimmed shell analysis.•It enables seamless and accurate large-deformation stress analysis, illustrated with a 76-patch model of a Dodge RAM hood.
Phase-field models based on the variational formulation for brittle fracture have recently been shown capable of accurately and robustly predicting complex crack behavior. Their numerical ...implementation requires costly operations at the quadrature point level, which may include finding eigenvalues and forming tensor projection operators. We explore the application of isogeometric collocation methods for the discretization of second-order and fourth-order phase-field fracture models. We show that a switch from isogeometric Galerkin to isogeometric collocation methods has the potential to significantly speed up phase-field fracture computations due to a reduction of point evaluations. We advocate a hybrid collocation–Galerkin formulation that provides a consistent way of weakly enforcing Neumann boundary conditions and multi-patch interface constraints, is able to handle the multiple boundary integral terms that arise from the weighted residual formulation, and offers the flexibility to adaptively improve the crack resolution in the fracture zone. We present numerical examples in one and two dimensions that illustrate the advantages of our approach.
•Isogeometric collocation can significantly speed up phase-field fracture computations.•We advocate a hybrid collocation–Galerkin formulation.•It handles Neumann boundary and multi-patch conditions, and higher-order boundary terms.•The adaptive Galerkin resolution of the fracture zone is crucial for accuracy and efficiency.
Objective: The branching behavior of vascular trees is often characterized using Murray's law. We investigate its validity using synthetic vascular trees generated under global optimization criteria. ...Methods: Our synthetic tree model does not incorporate Murray's law explicitly. Instead, we show that its validity depends on properties of the optimization model and investigate the effects of different physical constraints and optimization goals on the branching exponent that is now allowed to vary locally. In particular, we include variable blood viscosity due to the Fåhræus-Lindqvist effect and enforce an equal pressure drop between inflow and the micro-circulation. Using our global optimization framework, we generate vascular trees with over one million terminal vessels and compare them against a detailed corrosion cast of the portal venous tree of a human liver. Results: Murray's law is fulfilled when no additional constraints are enforced, indicating its validity in this setting. Variable blood viscosity or equal pressure drop lead to different optima but with the branching exponent inside the experimentally predicted range between 2.0 and 3.0. The validation against the corrosion cast shows good agreement from the portal vein down to the venules. Conclusion: Not enforcing Murray's law increases the predictive capabilities of synthetic vascular trees, and in addition reduces the computational cost. Significance: The ability to study optimal branching exponents across different scales can improve the functional assessment of organs.
We explore the use of various element-based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis. The rules studied encompass ...tensor-product Gauss and Gauss–Lobatto rules, and certain so-called monomial rules that do no possess a tensor-product structure. The objective of the study is to determine quadrature strategies, which enjoy the same accuracy and stability behavior as full Gauss quadrature, but with significantly fewer quadrature points. Several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard C0-continuous finite elements of the same order.
•We test various element-based reduced quadrature rules for quadratic and cubic spline elements.•They encompass tensor-product Gauss and Gauss–Lobatto rules, and monomial rules.•Some rules enjoy the same accuracy and stability as full Gauss quadrature, but with significantly fewer quadrature points.•They can substantially reduce the formation and assembly effort in isogeometric analysis.
► Hierarchical refinement of NURBS offers full analysis suitability, straightforward implementation and simple generalization to 3D. ► We first explore local hierarchical refinement for adaptive ...NURBS-based IGA. ► We then combine the B-spline version of the FCM and hierarchical refinement for a seamless design-through-analysis integration of 3D T-spline surface based models.
We explore hierarchical refinement of NURBS as a basis for adaptive isogeometric and immersed boundary analysis. We use the principle of B-spline subdivision to derive a local refinement procedure, which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions. We test hierarchical refinement of NURBS for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases. Using the B-spline version of the finite cell method, we illustrate the potential of immersed boundary methods as a seamless isogeometric design-through-analysis procedure for complex engineering parts defined by T-spline CAD surfaces, specifically a ship propeller and an automobile wheel. We show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features.
In this paper, we introduce a new framework for generating synthetic vascular trees, based on rigorous model-based mathematical optimization. Our main contribution is the reformulation of finding the ...optimal global tree geometry into a nonlinear optimization problem (NLP). This rigorous mathematical formulation accommodates efficient solution algorithms such as the interior point method and allows us to easily change boundary conditions and constraints applied to the tree. Moreover, it creates trifurcations in addition to bifurcations. A second contribution is the addition of an optimization stage for the tree topology. Here, we combine constrained constructive optimization (CCO) with a heuristic approach to search among possible tree topologies. We combine the NLP formulation and the topology optimization into a single algorithmic approach. Finally, we attempt the validation of our new model-based optimization framework using a detailed corrosion cast of a human liver, which allows a quantitative comparison of the synthetic tree structure with the tree structure determined experimentally down to the fifth generation. The results show that our new framework is capable of generating asymmetric synthetic trees that match the available physiological corrosion cast data better than trees generated by the standard CCO approach.
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