The aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with ...Rubin and Graves linearizing technique has been used. Modified cubic B‐spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate numerical algorithm is obtained. Three main test problems are solved for various values of the coefficients. To observe the performance of the present method, the error norms of the single soliton problem which has analytical solution are calculated. Besides these error norms, three invariants are reported. Comparison of the results displays that our algorithm produces superior results than those given in the literature.
Isogeometric analysis using LR B-splines Johannessen, Kjetil André; Kvamsdal, Trond; Dokken, Tor
Computer methods in applied mechanics and engineering,
02/2014, Letnik:
269
Journal Article
Recenzirano
Odprti dostop
The recently proposed locally refined B-splines, denoted LR B-splines, by Dokken et al. (2013) 6 may have the potential to be a framework for isogeometric analysis to enable future interoperable ...computer aided design and finite element analysis. In this paper, we propose local refinement strategies for adaptive isogeometric analysis using LR B-splines and investigate its performance by doing numerical tests on well known benchmark cases. The theory behind LR B-spline is not presented in full details, but the main conceptual ingredients are explained and illustrated by a number of examples.
We present a robust and efficient multigrid method for single-patch isogeometric discretizations using tensor product B-splines of maximum smoothness. Our method is based on a stable splitting of the ...spline space into a large subspace of 'Interior" splines which satisfy a robust inverse inequality, as well as one or several smaller subspaces which capture the boundary effects responsible for the spectral outliers which occur in isogeometric analysis. We then construct a multigrid smoother based on an additive subspace correction approach, applying a different smoother to each of the subspaces. For the interior splines, we use a mass smoother, whereas the remaining components are treated with suitably chosen Kronecker product smoothers or direct solvers. We prove that the resulting multigrid method exhibits iteration numbers which are robust with respect to the spline degree and the mesh size. Furthermore, it can be efficiently realized for discretizations of problems in arbitrarily high geometric dimension. Some numerical examples illustrate the theoretical results and show that the iteration numbers also scale relatively mildly with the problem dimension.
We initiate the study of efficient quadrature rules for NURBS-based isogeometric analysis. A rule of thumb emerges, the “half-point rule”, indicating that optimal rules involve a number of points ...roughly equal to half the number of degrees-of-freedom, or equivalently half the number of basis functions of the space under consideration. The half-point rule is independent of the polynomial order of the basis. Efficient rules require taking into account the precise smoothness of basis functions across element boundaries. Several rules of practical interest are obtained, and a numerical procedure for determining efficient rules is presented.
We compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics. The new rules represent improvements over those used previously in isogeometric analysis.
Powell–Sabin B‐splines are enjoying an increased use in the analysis of solids and fluids, including fracture propagation. However, the Powell–Sabin B‐spline interpolation does not hold the Kronecker ...delta property and, therefore, the imposition of Dirichlet boundary conditions is not as straightforward as for the standard finite elements. Herein, we discuss the applicability of various approaches developed to date for the weak imposition of Dirichlet boundary conditions in analyses which employ Powell–Sabin B‐splines. We take elasticity and fracture propagation using phase‐field modeling as a benchmark problem. We first succinctly recapitulate the phase‐field model for propagation of brittle fracture, which encapsulates linear elasticity, and its discretization using Powell–Sabin B‐splines. As baseline solution we impose Dirichlet boundary conditions in a strong sense, and use this to benchmark the Lagrange multiplier, penalty, and Nitsche's methods, as well as methods based on the Hellinger‐Reissner principle, and the linked Lagrange multiplier method and its modified version.
In this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. ...The presence of the small diffusion parameter ϵ leads to a parabolic right boundary layer. A collocation method consisting of cubic B‐spline basis functions on an appropriate piecewise‐uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter‐uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds.
As individuals age, death is a competing risk for Alzheimer's disease (AD) but the reverse is not the case. As such, studies of AD can be placed within the semi‐competing risks framework. Central to ...semi‐competing risks, and in contrast to standard competing risks , is that one can learn about the dependence structure between the two events. To‐date, however, most methods for semi‐competing risks treat dependence as a nuisance and not a potential source of new clinical knowledge. We propose a novel regression‐based framework that views the two time‐to‐event outcomes through the lens of a longitudinal bivariate process on a partition of the time scales of the two events. A key innovation of the framework is that dependence is represented in two distinct forms, local and global dependence, both of which have intuitive clinical interpretations. Estimation and inference are performed via penalized maximum likelihood, and can accommodate right censoring, left truncation, and time‐varying covariates. An important consequence of the partitioning of the time scale is that an ambiguity regarding the specific form of the likelihood contribution may arise; a strategy for sensitivity analyses regarding this issue is described. The framework is then used to investigate the role of gender and having ≥1 apolipoprotein E (APOE) ε4 allele on the joint risk of AD and death using data from the Adult Changes in Thought study.
In the present study, the coupled Burgers equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which cubic trigonometric and quintic ...B‐splines are used as approximate functions. In order to support the present study, three test problems given with appropriate initial and boundary conditions are going to be investigated. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the error norms
L2$$ {L}_2 $$ and
L∞$$ {L}_{\infty } $$. A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.
Summary
The analysis of (dynamic) fracture often requires multiple changes to the discretisation during crack propagation. The state vector from the previous time step must then be transferred to ...provide the initial values of the next time step. A novel methodology based on a least‐squares fit is proposed for this mapping. The energy balance is taken as a constraint in the mapping, which results in a complete energy preservation. Apart from capturing the physics better, this also has advantages for numerical stability. To further improve the accuracy, Powell‐Sabin B‐splines, which are based on triangles, have been used for the discretisation. Since
C1 continuity of the displacement field holds at crack tips for Powell‐Sabin B‐splines, the stresses at and around crack tips are captured much more accurately than when using elements with a standard Lagrangian interpolation, or with NURBS and T‐splines. The versatility and accuracy of the approach to simulate dynamic crack propagation are assessed in two case studies, featuring mode‐I and mixed‐mode crack propagation.
Summary
This paper presents an isogeometric collocation method for a computationally expedient random field discretization by means of the Karhunen‐Loève expansion. The method involves a collocation ...projection onto a finite‐dimensional subspace of continuous functions over a bounded domain, basis splines (B‐splines) and nonuniform rational B‐splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Similar to the existing Galerkin isogeometric method, the isogeometric collocation method preserves an exact geometrical representation of many commonly used physical or computational domains and exploits the regularity of isogeometric basis functions delivering globally smooth eigensolutions. However, in the collocation method, the construction of the system matrices for a d‐dimensional eigenvalue problem asks for at most d‐dimensional domain integrations, as compared with 2d‐dimensional integrations required in the Galerkin method. Therefore, the introduction of the collocation method for random field discretization offers a huge computational advantage over the existing Galerkin method. Three numerical examples, including a three‐dimensional random field discretization problem, illustrate the accuracy and convergence properties of the collocation method for obtaining eigensolutions.
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