GPU parallel strategy for parameterized LSM-based topology optimization using isogeometric analysis

This paper proposes a new level set-based topology optimization (TO) method using a parallel strategy of Graphics Processing Units (GPUs) and the isogeometric analysis (IGA). The strategy consists of parallel implementations for initial design domain, IGA, sensitivity analysis and design variable update, and the key issues in the parallel implementation, e.g., the parallel assembly race condition, are discussed in detail. The computational complexity and parallelization of the different steps in the TO are also analyzed in this paper. To better demonstrate the advantages of the proposed strategy, we compare efficiency of serial CPU, multi-thread parallel CPU and GPU by benchmark examples, and the speedups achieve two orders of magnitude.     

Efficient quadrature for NURBS-based isogeometric analysis

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.     

Robust multigrid solvers for the biharmonic problem in isogeometric analysis

In this paper, we develop multigrid solvers for the biharmonic problem in the framework of isogeometric analysis (IgA). In this framework, one typically sets up B-splines on the unit square or cube and transforms them to the domain of interest by a global smooth geometry function. With this approach, it is feasible to set up $H^2$-conforming discretizations. We propose two multigrid methods for such a discretization, one based on Gauss–Seidel smoothing and one based on mass smoothing. We prove that both are robust in the grid size, the latter is also robust in the spline degree. Numerical experiments illustrate the convergence theory and indicate the efficiency of the proposed multigrid approaches, particularly of a hybrid approach combining both smoothers.     

Compositional analysis for verification of parameterized systems

Many safety-critical systems that have been considered by the verification community are parameterized by the number of concurrent components in the system, and hence describe an infinite family of systems. Traditional model checking techniques can only be used to verify specific instances of this family. In this paper, we present a technique based on compositional model checking and program analysis for automatic verification of infinite families of systems. The technique views a parameterized system as an expression in a process algebra (CCS) and interprets this expression over a domain of formulas (modal mu-calculus), considering a process as a property transformer. The transformers are constructed using partial model checking techniques. At its core, our technique solves the verification problem by finding the limit of a chain of formulas. We present a widening operation to find such a limit for properties expressible in a subset of modal mu-calculus. We describe the verification of a number of parameterized systems using our technique to demonstrate its utility.     

On parameterized matrix splitting preconditioner for the saddle point problems

In this paper, we present a parameterized matrix splitting (PMS) preconditioner for the large sparse saddle point problems. The preconditioner is based on a parameterized splitting of the saddle point matrix, resulting in a fixed-point iteration. The convergence theorem of the new iteration method for solving large sparse saddle point problems is proposed by giving the restrictions imposed on the parameter. Based on the idea of the parameterized splitting, we further propose a modified PMS preconditioner. Some useful properties of the preconditioned matrix are established. Numerical implementations show that the resulting preconditioner leads to fast convergence when it is used to precondition Krylov subspace iteration methods such as generalized minimal residual method.     

Generalized B-splines as a tool in isogeometric analysis

The concept of isogeometric analysis has been proposed in [13], where NURBS are considered as basis of the analysis, thanks to their ability to construct an exact geometric model in several practical applications and to their popularity in commercial CAD systems. In this paper we propose an alternative to the rational model presenting an isogeometric analysis approach based on generalized B-splines. Geometric models exactly represented by generalized B-splines include those generated by NURBS. Moreover, generalized B-splines possess all fundamental properties of algebraic B-splines (and NURBS) including classical refinement processes as h–p–k refinements. Finally, since generalized B-splines are not confined to rational functions, they behave completely similar to algebraic B-splines with respect to differentiation and integration. This seems to be of interest in the treatment of some relevant problems.     

Analytic formulation of stiffness matrix for axisymmetric solids

Stiffness matrices for axisymmetric solids with arbitrary loading were derived through the application of variational principles in analytic form. The singularities in the stiffness matrices were removed through displacement constraints along the axis of symmetry for each circumferential mode. The shear stresses and maximum deflections of a set of Saint-Venant flexural problems were obtained both analytically and numerically. The results indicate that the finite element analysis with the analytic stiffness matrix provides a very good solution. The same problems were solved with a commercial code, ANSYS. and showed that the analytic stiffness matrix contributed to a faster convergence rate as the number of elements increases in an analysis.     

Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis

Isogeometric analysis has been proposed as a methodology for bridging the gap between computer aided design (CAD) and finite element analysis (FEA). Although both the traditional and isogeometric pipelines rely upon the same conceptualization to solid model steps, they drastically differ in how they bring the solid model both to and through the analysis process. The isogeometric analysis process circumvents many of the meshing pitfalls experienced by the traditional pipeline by working directly within the approximation spaces used by the model representation. In this paper, we demonstrate that in a similar way as how mesh quality is used in traditional FEA to help characterize the impact of the mesh on analysis, an analogous concept of model quality exists within isogeometric analysis. The consequence of these observations is the need for a new area within modeling – analysis-aware modeling – in which model properties and parameters are selected to facilitate isogeometric analysis.     

Exact stiffness matrix for beams on elastic foundation

An exact stiffness matrix of a beam element on elastic foundation is formulated. A single element is required to exactly represent a continuous part of a beam on a Winkler foundation. Thus only a few elements are sufficient for a typical problem solution. The stiffness matrix is assembled in a computer program and some numerical examples are presented.     

A matrix gradient algorithm for identification of parameterized time‐varying parameters

This paper considers the problem of estimating time‐varying parameters which can be parameterized by a series of arbitrary known basis functions. It is shown that this problem is equivalent to the observer design problem for a “matrix” dynamic system. A “matrix” gradient algorithm, which mimics the well‐known “vector” gradient algorithm, is proposed to estimate the unknown matrix. The contribution of this paper is to show that convergence of the proposed matrix algorithm is guaranteed by the persistent excitations of both the regressor and the basis functions. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

H2 regularity properties of singular parameterizations in isogeometric analysis

Isogeometric analysis (IGA) is a numerical simulation method which is directly based on the NURBS-based representation of CAD models. It exploits the tensor-product structure of 2- or 3-dimensional NURBS objects to parameterize the physical domain. Hence the physical domain is parameterized with respect to a rectangle or to a cube. Consequently, singularly parameterized NURBS surfaces and NURBS volumes are needed in order to represent non-quadrangular or non-hexahedral domains without splitting, thereby producing a very compact and convenient representation.The Galerkin projection introduces finite-dimensional spaces of test functions in the weak formulation of partial differential equations. In particular, the test functions used in isogeometric analysis are obtained by composing the inverse of the domain parameterization with the NURBS basis functions. In the case of singular parameterizations, however, some of the resulting test functions do not necessarily fulfill the required regularity properties. Consequently, numerical methods for the solution of partial differential equations cannot be applied properly.We discuss the regularity properties of the test functions. For one- and two-dimensional domains we consider several important classes of singularities of NURBS parameterizations. For specific cases we derive additional conditions which guarantee the regularity of the test functions. In addition we present a modification scheme for the discretized function space in case of insufficient regularity. It is also shown how these results can be applied for computational domains in higher dimensions that can be parameterized via sweeping.     

A hierarchical approach to adaptive local refinement in isogeometric analysis

Adaptive local refinement is one of the key issues in isogeometric analysis. In this article we present an adaptive local refinement technique for isogeometric analysis based on extensions of hierarchical B-splines. We investigate the theoretical properties of the spline space to ensure fundamental properties like linear independence and partition of unity. Furthermore, we use concepts well-established in finite element analysis to fully integrate hierarchical spline spaces into the isogeometric setting. This also allows us to access a posteriori error estimation techniques. Numerical results for several different examples are given and they turn out to be very promising.     

Support-free robust topology optimization based on pseudo-inverse stiffness matrix and eigenvalue analysis

Structural and Multidisciplinary Optimization - Current finite element analysis (FEA) and optimizations require boundary conditions, i.e., constrained nodes. These nodes represent structural...     

Symbolic reachability analysis for parameterized administrative role-based access control

Role-based access control (RBAC) is a widely used access control paradigm. In large organizations, the RBAC policy is managed by multiple administrators. An administrative role-based access control (ARBAC) policy specifies how each administrator may change the RBAC policy. It is often difficult to fully understand the effect of an ARBAC policy by simple inspection, because sequences of changes by different administrators may interact in unexpected ways. ARBAC policy analysis algorithms can help by answering questions, such as user-role reachability, which asks whether a given user can be assigned to given roles by given administrators.Allowing roles and permissions to have parameters significantly enhances the scalability, flexibility, and expressiveness of ARBAC policies. This paper defines PARBAC, which extends the classic ARBAC97 model to support parameters, proves that user-role reachability analysis for PARBAC is undecidable when parameters may range over infinite types, and presents a semi-decision procedure for reachability analysis of PARBAC. To the best of our knowledge, this is the first analysis algorithm specifically for parameterized ARBAC policies. We evaluate its efficiency by analyzing its parameterized complexity and benchmarking it on case studies and synthetic policies. We also experimentally evaluate the effectiveness of several optimizations.     

High-performance computing of wind turbine aerodynamics using isogeometric analysis

In this article we present a high-performance computing framework for advanced flow simulation and its application to wind energy based on the residual-based variational multiscale (RBVMS) method and isogeometric analysis. The RBVMS formulation and its suitability and accuracy for turbulent flow in a moving domain are presented. Particular emphasis is placed on the parallel implementation of the methodology and its scalability. Two challenging flow cases were considered: the turbulent Taylor–Couette flow and the NREL 5 MW offshore baseline wind turbine rotor at full scale. In both cases, flow quantities of interest from the simulation results compare favorably with the reference data and near-perfect linear parallel scaling is achieved.     

Parameterization of computational domain in isogeometric analysis: Methods and comparison

Parameterization of computational domain plays an important role in isogeometric analysis as mesh generation in finite element analysis. In this paper, we investigate this problem in the 2D case, i.e., how to parametrize the computational domains by planar B-spline surface from the given CAD objects (four boundary planar B-spline curves). Firstly, two kinds of sufficient conditions for injective B-spline parameterization are derived with respect to the control points. Then we show how to find good parameterization of computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of planar B-spline parameterization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of planar B-spline parameterization. By using this method, the resulted parameterization has no self-intersections, and the isoparametric net has good uniformity and orthogonality. After introducing a posteriori error estimation for isogeometric analysis, we propose r-refinement method to optimize the parameterization by repositioning the inner control points such that the estimated error is minimized. Several examples are tested on isogeometric heat conduction problem to show the effectiveness of the proposed methods and the impact of the parameterization on the quality of the approximation solution. Comparison examples with known exact solutions are also presented.     

Sensitivity analysis of singularly perturbed systems

A new variational approach is described to compute singular sensitivity functions of finite dimensional systems with respect to changes in system structure. The reduced nominal model and its adjoint system are used to define singular sensitivity functions. A relation between the present method and singularly perturbed optimal control theory is established. It is shown that the present variational approach gives a computationally effective method in singular sensitivity analysis. The method is applied to singularly perturbed linear ordinary differential equations and singular optimal control problem.