Isogeometric Collocation Method Research Articles

Isogeometric collocation method to simulate phase-field crystal model

Isogeometric collocation method to simulate phase-field crystal model

An extended isogeometric collocation method for fracture analysis

AbstractA collocation method is developed for discrete fracture models in the context of the partition‐of‐unity method. Spline technologies used in isogeometric analysis (IGA) are exploited to provide a smooth inter‐element transition of gradients, thus allowing to get rid of extra flux terms at element boundaries which are generated by Lagrange polynomials. Bézier extraction is utilised to formulate IGA commensurate with a standard finite element data‐structure. The efficacy of the proposed approach is examined through different numerical examples and is compared with other discrete methods for fracture analysis. The proposed approach is competitive in terms of accuracy with the least computational cost, rendering it a suitable candidate for superseding available collocation approaches for fracture simulation. Moreover, the approach naturally assesses the possibility of physics informed neural networks for fracture simulation, to which collocation is central.

Stabilized isogeometric collocation methods for hyperbolic conservation laws

Stabilized isogeometric collocation methods for hyperbolic conservation laws

Enhanced domain decomposition Schwarz solution schemes for isogeometric collocation methods

Isogeometric collocation methods have been introduced as an alternative to isogeometric Galerkin formulations, aiming at improving the computational cost of simulation by reducing the cost of assembly of the corresponding matrices. However, in contrast to their Galerkin counterparts, collocation formulations result in non-symmetric matrices of much higher dimensions, for a specified level of accuracy, which require special attention when addressing large-scale simulations. Furthermore, the presence of mixed boundary conditions may lead to indefinite collocation matrices, which hinder the convergence properties of domain decomposition-based iterative solution methods of the corresponding algebraic equations. To address these shortcomings, two-level hybrid, domain decomposition-based preconditioners are proposed, related to both the basic and the enhanced collocation formulations, in conjunction with the generalized minimal residual method and Schwarz-based preconditioners. The proposed solution schemes improve the computational efficiency of the isogeometric collocation simulations and exhibit numerical scalability for an increased number of subdomains. This is attributed to the fact that the iterations are performed on the Schur complement level of the corresponding matrices, leading to stable, computationally efficient, and robust solutions of the corresponding non-symmetric algebraic equations.

An isogeometric collocation method for the static limit analysis of masonry domes under their self-weight

A novel isogeometric collocation method is proposed for the static limit analysis of axially-symmetric masonry domes subject to their self-weight. A shell-based static formulation is employed, alongside Heyman’s assumptions on masonry, to characterize the equilibrated and statically admissible stress states in the dome. As a distinctive feature of the approach, a vector stress function is introduced, generating point-wise self-equilibrated shell stress resultants in the dome. Accordingly, the classical minimum-thrust problem is formulated in terms of the unknown vector stress function, and the static admissibility conditions are enforced as the only optimization constraint. NURBS-based isogeometric analysis is adopted to accomplish the need for an accurate geometric description of the dome and a high-order continuous interpolation of the vector stress function. A discrete minimum-thrust problem is derived as a linear programming problem, with the static admissibility conditions checked at suitable collocation points. Instrumental to its solution is the computation of a particular solution of the equilibrium equations, which is obtained by an isogeometric collocation method. By a mechanical interpretation of the dual optimization problem, the settlement mechanism of the dome corresponding to its minimum-thrust state is also computed. Numerical results, dealing with a thorough convergence analysis, parametric analyses on spherical and ogival domes with parameterized geometry, and the real case of the Taj Mahal central dome are presented to prove the computational merit of the proposed approach.

Stabilized isogeometric collocation methods for scalar transport and incompressible fluid flow

In this work we adapt classical residual-based stabilization techniques to the spline collocation setting. Inspired by the Streamline-Upwind-Petrov-Galerkin and Pressure-Stabilizing-Petrov-Galerkin methods, our stabilized collocation schemes address spurious oscillations that can arise from advection and pressure instabilities. Numerical examples for the advection–diffusion equation, Stokes equations, and incompressible Navier–Stokes equations show the effectiveness of the proposed stabilized schemes while maintaining the high-order convergence rates and accuracy of standard isogeometric collocation on smooth problems.

Divergence-conforming isogeometric collocation methods for the incompressible Navier–Stokes equations

We develop two isogeometric divergence-conforming collocation schemes for incompressible flow. The first is based on the standard, velocity–pressure formulation of the Navier–Stokes equations, while the second is based on the rotational form and includes the vorticity as an unknown in addition to the velocity and pressure. We describe the process of discretizing each unknown using B-splines that conform to a discrete de Rham complex and collocating each governing equation at the Greville abcissae corresponding to each discrete space. Results on mapped domains are obtained by mapping the equations back to a parametric domain using structure-preserving transformations. Numerical results show the promise of the method, including accelerated convergence rates of the three field, vorticity–velocity–pressure scheme when compared to the two field, velocity–pressure scheme.

IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method

The isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressive-iterative approximation (IG-LSPIA), to solve the fitting problem in the collocation method. IG-LSPIA starts with an initial blending function, where the control coefficients are combined with the B-spline basis functions and their derivatives. A new blending function is generated by constructing the differences for collocation points (DCP) and control coefficients (DCC), and then adding the DCC to the corresponding control coefficients. The procedure is performed iteratively until the stop criterion is reached. We prove the convergence of IG-LSPIA and show that the computation complexity in each iteration of IG-LSPIA is related only to the number of collocation points and unrelated to the number of control coefficients. Moreover, an incremental algorithm is designed; it alternates with knot refinement until the desired precision is achieved. After each knot refinement, the result of the last round of IG-LSPIA iterations is used to generate the initial blending function of the new round of iteration, thereby saving great computation. Experiments show that the proposed method is stable and efficient. In the three-dimensional case, the total computation time is saved twice compared to the traditional method.

A Survey on Isogeometric Collocation Methods with Applications

Isogeometric analysis (IGA) is an effective numerical method for connecting computer-aided design and engineering, which has been widely applied in various aspects of computational mechanics. IGA involves Galerkin and collocation formulations. Exploiting the same high-order non-uniform rational B-spline (NURBS) bases that span the physical domain and the solution space leads to increased accuracy and fast computation. Although IGA Galerkin provides optimal convergence, IGA collocation performs better in terms of the ratio of accuracy to computational time. Without numerical integration, by working directly with the strong form of the partial differential equation over the physical domain defined by NURBS geometry, the derivatives of the NURBS-expressed numerical solution at some chosen collocation points can be calculated. In this study, we survey the methodological framework and the research prospects of IGA. The collocation schemes in the IGA collocation method that affect the convergence performance are addressed in this paper. Recent studies and application developments are reviewed as well.

Immersed isogeometric analysis based on a hybrid collocation/finite cell method

We propose an immersed approach for the isogeometric collocation method, combined with the Galerkin-based finite cell method, to avoid the subdivision of complex geometries in too many patches.The presented technology further develops the hybrid collocation concept to accommodate both numerical methods within a single framework, providing a systematic technique for selecting the method to be used.We perform several numerical tests to demonstrate that the methodology guarantees the same convergence rates obtained using the standard isogeometric collocation method.

An improved isogeometric collocation formulation for spatial multi-patch shear-deformable beams with arbitrary initial curvature

The present paper presents a robust multi-patch formulation based on the isogeometric collocation (IGA-C) method for the solution of linear, spatial Timoshenko beam structures with complex geometry. The proposed approach is based on the combination of the (local) Bishop frame with the exponential map for SO(3) (Rodrigues formula) to compute the beam curvature and its derivative. This choice permits bypassing known issues related to the Serret–Frenet (SF) frame (in cases where the local beam curvature vanishes) and does not require the Darboux vector and its derivative, which are both affected by limitations of the SF frame. Moreover, in contrast to existing formulations mostly derived in the local SF frame, here the formulation is consistently derived by linearizing the governing equations in the material setting, where the multi-patch coupling can be enforced in a straightforward way. Numerical tests are performed on complex spatial curved beams, including a demanding biomechanical problem of a braided stent, proving the accuracy and the robustness of the formulation.

Isogeometric collocation method based on residual parameterization of planar physical domain

Isogeometric analysis (IGA) becomes an effective tool for solving partial differential equations (PDEs). Compared with isogeometric Galerkin method, isogeometric collocation (IGC) method has higher computational efficiency. In this paper, we aim to optimize the domain parameterization in IGC for better numerical accuracy of solving PDEs. Firstly, we present a new parameterization method of planar physical domain, called residual parameterization, which is obtained by minimizing the objective functions consisting of geometry-related functionals and the analysis-related residual norms in an unconstrained optimization problem. Secondly, the reduced quadrature rules are applied to residual norms due to high computational cost in evaluating the integration of residual terms. Finally, based on the residual parameterization, we solve boundary value problems (BVPs) by IGC with Greville points and superconvergent points to verify the strength of our proposed residual parameterization of planar physical domain. Several numerical examples show that the numerical accuracy of the proposed method is improved nearly half order of magnitude compared with standard IGC method.

Mixed isogeometric collocation for geometrically exact 3D beams with elasto-visco-plastic material behavior and softening effects

A geometrically nonlinear, shear-deformable 3D beam formulation with inelastic material behavior and its numerical discretization by a mixed isogeometric collocation method are presented. In particular, the constitutive model captures elasto-visco-plasticity with damage/softening from Mullin’s effect, which applies to the modeling of metallic and polymeric materials, e.g., in additive manufacturing applications and metamaterials. The inelastic material behavior is formulated in terms of thermodynamically consistent internal variables for viscoelastic and plastic strains and isotropic and kinematic hardening variables, as well as accompanying evolution equations. A mixed isogeometric collocation method is applied for the discretization of the strong form of the quasi-static nonlinear differential equations. Thus, the displacements of the centerline curve, the cross-section orientations, and the stress resultants (forces and moments) are discretized as B-spline or NURBS curves. The internal variables are defined only locally at the collocation points, and an implicit return-mapping algorithm is employed for their time discretization. The method is verified in comparison to 1D examples as well as reference results for 3D beams. Furthermore, its applicability to the simulation of beam lattice structures subject to large deformations and instabilities is demonstrated.

Static, free vibrational and buckling analysis of laminated composite beams using isogeometric collocation method

Isogeometric collocation (IGA-C) method is a computational approach to solve boundary value problems. In this method (IGA-C), the differential equations are solved in strong form instead of the weak form approach adopted by Galerkin based formulations. IGA-C method is computationally efficient in comparison to conventional finite element method and Galerkin-Isogeometric approaches. IGA-C method does not involve the process of assembling global stiffness matrix from element stiffness matrix. Another advantage of IGA-C is that it requires a single integration point per element irrespective of the order of Non-Uniform Rational B-Spline functions (NURBS) adopted. Isogeometric collocation has also been demonstrated as a stable, efficient and accurate higher order computational method for explicit problems. For a wider adoption of isogeometric collocation method, beam/plate/shell finite elements within the framework of IGA-C method need to be formulated. Owing to the extensive adoption of laminated composites in structural components, development of beam finite elements for laminated composite beams based on isogeometric collocation method will prove useful during analysis of composite structures. IGA-C method is proposed in this study for the static bending, free vibration and buckling analysis of laminated composite beams. Classical laminated plate theory (CLPT), first order shear deformation theory (FSDT) and higher order shear deformation theory (HSDT) are considered for all the three analyses. The computational approach proposed for laminated beam based on HSDT contains two Degrees of Freedom (DOF) per node. Computational approach for analysing laminated composite beams based on each of these kinematic theories and using IGA-C method is presented. Accuracy of the proposed computational approaches is checked by solving different numerical examples. Values of normalized transverse displacement, normalized stresses, normalized natural frequencies and normalized critical buckling loads are compared with results from the literature and are found to be accurate.

The convergence rate and necessary-and-sufficient condition for the consistency of isogeometric collocation method

Although the isogeometric collocation (IGA-C) method has been successfully utilized in practical applications due to its simplicity and efficiency, only a little theoretical results have been established on the numerical analysis of the IGA-C method. In this paper, we deduce the convergence rate of the consistency of the IGA-C method. Moreover, based on the formula of the convergence rate, the necessary and sufficient condition for the consistency of the IGA-C method is developed. These results advance the numerical analysis of the IGA-C method.

Isogeometric sizing and shape optimization of 3D beams and lattice structures at large deformations

A computational method for optimizing the shape of the centerline curve and the spatial variation of geometric and material sizing parameters of the cross-sections of elastic, 3-dimensional beams and beam structures subject to large deformations is presented in this work. The approach is based on the concept of isogeometric analysis, i.e., the representation of geometry and the discretization of the numerical solution using spline functions. Here, mixed isogeometric collocation methods are used to discretize the geometrically exact 3D beam model. These spline representations are extended to the parameterization of the design variables, which are the initial centerline curves of the beams, as well as cross-sectional sizing properties, which may be varying along the beam axis and can be functionally graded through the cross-sections. To tailor the mechanical deformation behavior of a beam or beam structure, a nonlinear optimization problem is formulated and solved using gradient-based methods. For this purpose, all required gradients and sensitivities are derived analytically. The potential of this holistic design optimization approach is demonstrated in application to tailoring of elastic metamaterials and beam lattice structures, as well as 4D printing of multi-material laminate beams.

Isogeometric collocation method with intuitive derivative constraints for PDE-based analysis-suitable parameterizations

This paper presents a general formulation of an isogeometric collocation method (IGA-C) for the parameterization of computational domains for the isogeometric analysis (IGA) using non-uniform rational B-splines (NURBS). The boundary information of desired computational domains for IGA is imposed as a Dirichlet boundary condition on a simple and smooth initial parameterization of an initial computational domain, and the final parameterization is produced based on the numerical solution of a partial differential equation (PDE) that is solved using the IGA-C method. In addition, we apply intuitive derivative constraints while solving the PDE to achieve desired properties of smoothness and uniformity of the resulting parameterization. While one may use any general PDE with any constraint, the PDEs and additional constraints selected in our case are such that the resulting solution can be efficiently solved through a system of linear equations with or without additional linear constraints. This approach is different from typical existing parameterization methods in IGA that are often solved through an expensive nonlinear optimization process. The results show that the proposed method can efficiently produce satisfactory analysis-suitable parameterizations.

Overlapping Additive Schwarz preconditioners for isogeometric collocation discretizations of linear elasticity

Overlapping Additive Schwarz (OAS) preconditioners are here constructed for isogeometric collocation discretizations of the system of linear elasticity in both two and three space dimensions. Isogeometric collocation methods are recent variants of isogeometric analysis based on the numerical approximation of the strong form of partial differential equations at appropriate collocation points. Numerical results in two and three dimensions show that two-level OAS preconditioners are scalable in the number of subdomains N, quasi-optimal with respect to the mesh size h and optimal with respect to the spline polynomial degree p. Moreover, two-level OAS preconditioners are more robust than one-level OAS and non-preconditioned GMRES solvers when the material tends to the incompressible limit, as well as in the presence of strong deformation of the NURBS geometry.

Superconvergent isogeometric collocation method with Greville points

A superconvergent isogeometric collocation method that employs the conventional Greville abscissae as collocation points is proposed. Firstly, a basis transformation is presented for isogeometric basis functions, which preserves the exact geometry representation characterized by isogeometric analysis. More specifically, the geometry is exactly represented by the transformed isogeometric basis functions and the Greville points that function as control points in the current formulation. Meanwhile, the proposed basis transformation enables a very straightforward formulation of the consistency conditions. It also turns out that the transformed isogeometric basis functions span the same solution space as the standard isogeometric basis functions. Subsequently, with the aid of these consistency conditions, an accuracy analysis is presented for multi-dimensional isogeometric collocation formulation and the basis degree discrepancy issue is theoretically identified. In order to resolve this issue, a recursive gradient formulation is further introduced to effectively construct the gradients of transformed isogeometric basis functions. It is shown that these recursive gradients meet additional higher order consistency conditions. Consequently, a superconvergent isogeometric collocation method is finally established, which utilizes the recursive gradients to replace the conventional gradients of isogeometric basis functions and directly collocates at Greville points. The superconvergence of the proposed method regarding odd degree basis functions is then proved, which is also well manifested by numerical results.

Isogeometric Collocation: A Mixed Displacement-Pressure Method for Nearly Incompressible Elasticity

We investigate primal and mixed u−p isogeometric collocation methods for application to nearly-incompressible isotropic elasticity. The primal method employs Navier’s equations in terms of the displacement unknowns, and ... | Find, read and cite all the research you need on Tech Science Press