An isogeometric analysis approach to gradient damage models Verhoosel, Clemens V.; Scott, Michael A.; Hughes, Thomas J. R. ...
International journal for numerical methods in engineering,
8 April 2011, Letnik:
86, Številka:
1
Journal Article
Isogeometric collocation methods have been proposed recently and their accuracy and efficiency demonstrated for elastostatics and explicit dynamics. This paper addresses two important aspects in the ...development of the isogeometric collocation technology, namely, the imposition of Neumann boundary conditions and the enforcement of contact constraints, which are both treated within the same framework. It is shown that the strong imposition of Neumann boundary conditions may lead to a significant loss of accuracy in some situations, in particular when non-uniform meshes are used. Two possible remedies are proposed to restore the desired level of accuracy while keeping the computational cost virtually unchanged, i.e. a hybrid collocation–Galerkin approach and an enhanced collocation (EC) approach. A frictionless contact formulation suitable for the collocation framework is further proposed and shown to pass the contact patch test to machine precision. When combined with the EC approach, the formulation is shown to deliver accurate results and to perform robustly also for highly non-uniform meshes. For all the collocation formulations, contact pressures are greater than or equal to zero pointwise, in contrast with standard Lagrange finite elements.
•Neumann boundary conditions in isogeometric collocation are discussed.•Two alternative formulations to standard isogeometric collocation are proposed.•The alternative formulations are more accurate than the standard one for non-uniform meshes.•A contact formulation for isogeometric collocation is proposed.•The contact formulation is shown to pass the contact patch test.
We develop a multi-degree polar spline framework with applications to both geometric modeling and isogeometric analysis. First, multi-degree splines are introduced as piecewise non-uniform rational ...B-splines (NURBS) of non-uniform or variable polynomial degree, and a simple algorithm for their construction is presented. Then, an extension to two-dimensional polar configurations is provided by means of a tensor-product construction with a collapsed edge. Suitable combinations of these basis functions, encoded in a so-called isogeometric analysis suitable extraction operator, yield Ck smooth polar splines for any k≥0. We show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality. Explicit constructions for k∈{0,1,2} are presented. Optimal approximation behavior is observed numerically, and examples of applications to free-form design, smooth hole-filling, and high-order partial differential equations demonstrate the applicability of the developed framework.
•A polar spline framework is proposed for geometric modeling and isogeometric analysis.•We present a practical construction for univariate splines of non-uniform degree.•This is extended to polar configurations using tensor-products with a collapsed edge.•A Ck smooth polar spline basis is developed that forms a convex partition of unity.•Optimal error convergence rates are observed for L2, H1 and H2 projection problems.
Abstract Background The importance of quality-of-life (QoL) research has been recognised over the past two decades in patients with head and neck (H&N) cancer. The aims of this systematic review are ...to evaluate the QoL status of H&N cancer survivors one year after treatment and to identify the determinants affecting their QoL. Methods Pubmed, Medline, Scopus, Sciencedirect and CINAHL (2000–2011) were searched for relevant studies, and two of the present authors assessed their methodological quality. The characteristics and main findings of the studies were extracted and reported. Results Thirty-seven studies met the inclusion criteria, and the methodological quality of the majority was moderate to high. While patients of the group in question recover their global QoL by 12 months after treatment, a number of outstanding issues persist – deterioration in physical functioning, fatigue, xerostomia and sticky saliva. Age, cancer site, stage of disease, social support, smoking, feeding tube placement and alcohol consumption are the significant determinants of QoL at 12 months, while gender has little or no influence. Conclusions Regular assessments should be carried out to monitor physical functioning, degree of fatigue, xerostomia and sticky saliva. Further research is required to develop appropriate and effective interventions to deal with these issues, and thus to promote the patients’ QoL.
We continue the study initiated in Hughes et al. (2010) in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis. These rules are optimal ...in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points. We extend the algorithm presented in Hughes et al. (2010) with an improved starting guess, which combined with arbitrary precision arithmetic, results in the practical computation of quadrature rules for univariate non-uniform splines up to any precision. Explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors. We study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two- and three-dimensional diffusion–reaction problems using tensor product and hierarchically refined splines, and prove a theorem rigorously establishing the stability and accuracy of the reduced rules.
► 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.
To achieve a tight integration of design and analysis, conformal solid T-spline construction with the input boundary spline representation preserved is desirable. However, to the best of our ...knowledge, this is still an open problem. In this paper, we provide its first solution. The input boundary T-spline surface has genus-zero topology and only contains eight extraordinary nodes, with an isoparametric line connecting each pair. One cube is adopted as the parametric domain for the solid T-spline. Starting from the cube with all the nodes on the input surface as T-junctions, we adaptively subdivide the domain based on the octree structure until each face or edge contains at most one face T-junction or one edge T-junction. Next, we insert two boundary layers between the input T-spline surface and the boundary of the subdivision result. Finally, knot intervals are calculated from the T-mesh and the solid T-spline is constructed. The obtained T-spline is conformal to the input T-spline surface with exactly the same boundary representation and continuity. For the interior region, the continuity is
C
2
everywhere except for the local region surrounding irregular nodes. Several examples are presented to demonstrate the performance of the algorithm.
This paper is devoted to the numerical simulation of the Navier–Stokes–Korteweg equations, a phase-field model for water/water-vapor two-phase flows. We develop a numerical formulation based on ...isogeometric analysis that permits straightforward treatment of the higher-order partial–differential operator that represents capillarity. We introduce a new refinement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions. Finally, we present several numerical examples in two and three dimensions that illustrate the effectiveness and robustness of our approach.
The outbreak of COVID-19 in 2020 has led to a surge in interest in the research of the mathematical modeling of epidemics. Many of the introduced models are so-called
compartmental models
, in which ...the total quantities characterizing a certain system may be decomposed into two (or more) species that are distributed into two (or more) homogeneous units called compartments. We propose herein a formulation of compartmental models based on partial differential equations (PDEs) based on concepts familiar to continuum mechanics, interpreting such models in terms of fundamental equations of balance and compatibility, joined by a constitutive relation. We believe that such an interpretation may be useful to aid understanding and interdisciplinary collaboration. We then proceed to focus on a compartmental PDE model of COVID-19 within the newly-introduced framework, beginning with a detailed derivation and explanation. We then analyze the model mathematically, presenting several results concerning its stability and sensitivity to different parameters. We conclude with a series of numerical simulations to support our findings.
Analysis-suitable T-splines (AST-splines) are a promising candidate to achieve a seamless integration between the design and the analysis of thin-walled structures in industrial settings. In this ...work, we allow multiple extraordinary points per face, i.e., we remove the restriction of preceding works that required extraordinary points to be at least four rings apart from each other. We do so by mathematically showing that AST-splines with multiple extraordinary points per face are linearly independent and their polynomial basis functions form a non-negative partition of unity. This extension of the subset of AST-splines drastically increases the flexibility to build geometries using AST-splines; e.g., much coarser meshes can be constructed around small holes. The AST-spline spaces detailed in this work have C1 inter-element continuity near extraordinary points and C2 inter-element continuity elsewhere. For the convergence studies performed in this paper involving second- and fourth-order linear elliptic problems with manufactured solutions, we have not found any drawback caused by allowing multiple EPs per face in either the first refinement levels or the asymptotic behavior. To illustrate a possible isogeometric framework that is already available, we design the B-pillar and the side outer panel of a car using T-splines with the commercial software Autodesk Fusion360, import the control nets into our in-house code to build AST-splines, and import the Bézier extraction information into the commercial software LS-DYNA to solve eigenvalue problems. The results are compared with conventional finite elements and good agreement is found between AST-splines and conventional finite elements.
•Smooth splines with multiple extraordinary points per face are studied.•A mathematical proof of linear independence is provided.•Excellent convergence from level 0 is obtained.•Geometries with arbitrary topological genus are built.•Comparisons with the commercial software LS-DYNA are included.
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