Non-uniform Rational B-spline Basis Functions Research Articles

A Fortran implementation of isogeometric analysis for thin plate problems with the penalty method

Purpose As one of the earliest high-level programming languages, Fortran with easy accessibility and computational efficiency is widely used in the engineering field. The purpose of this paper is to present a Fortran implementation of isogeometric analysis (IGA) for thin plate problems. Design/methodology/approach IGA based on non-uniform rational B-splines (NURBS) offers exact geometries and is more accurate than finite element analysis (FEA). Unlike the basis functions in FEA, NURBS basis functions are non-interpolated. Hence, the penalty method is used to enforce boundary conditions. Findings Several thin plate examples based on the Kirchhoff-Love theory were illustrated to demonstrate the accuracy of the implementation in contrast with analytical solutions, and the efficiency was validated in comparison with another open method. Originality/value A Fortran implementation of NURBS-based IGA was developed to solve Kirchhoff-Love plate problems. It easily obtained high-continuity basis functions, which are necessary for Kirchhoff formulation. In comparison with theoretical solutions, the numerical examples demonstrated higher accuracy and faster convergence of the Fortran implementation. The Fortran implementation can well solve the time-consuming problem, and it was validated by the time-consumption comparison with the Matlab implementation. Due to the non-interpolation of NURBS, the penalty method was used to impose boundary conditions. A suggestion of the selection of penalty coefficients was given.

Isogeometric Analysis of the Nonlinear Deformation of Planar Flexible Beams with Snap-back

Based on the continuity of the derivatives of the Non-Uniform Rational B-Splines (NURBS) curve and the Jaumann strain measure, the present paper adopted the position coordinates of the control points as the degrees of freedom and developed a planar rotation-free Euler-Bernoulli beam element for isogeometric analysis, where the derivatives of the field variables with respect to the arc-length were expressed as the sum of the weighted sum of the position coordinates of the control points, and the NURBS basis functions were used as the weight functions. Furthermore, the concept of bending strip was used to involve the rigid connection between multiple patches. Several typical examples with geometric nonlinearities were used to demonstrate the accuracy and effectiveness of the proposed algorithm. The presented formulation fully accounts for the geometric nonlinearities and can be used to study the snap-through and snap-back phenomena of flexible beams.

Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth

We present a novel numerical method to simulate crack growth in 3D, directly from the Computer-Aided Design (CAD) geometry of the component, without any mesh generation. The method is an isogeometric boundary element method (IGABEM) based on non-uniform rational B-splines (NURBS). NURBS basis functions are used for the domain and crack representation as well as to approximate the physical quantities involved in the simulations. A stable quadrature scheme for singular integration is proposed to enhance the robustness of the method in dealing with highly distorted elements. Convergence studies in the crack opening displacement is performed for a penny-shaped crack and an elliptical crack. Two approaches to extract stress intensity factors (SIFs): the contour M integral and the virtual crack closure integral are compared using dual integral equations. The results show remarkable accuracy in the computed SIFs, leading to smooth crack paths and reliable fatigue lives, without requiring the generation of any mesh from the CAD model of the component under consideration.

PetIGA: A framework for high-performance isogeometric analysis

We present PetIGA, a code framework to approximate the solution of partial differential equations using isogeometric analysis. PetIGA can be used to assemble matrices and vectors which come from a Galerkin weak form, discretized with Non-Uniform Rational B-spline basis functions. We base our framework on PETSc, a high-performance library for the scalable solution of partial differential equations, which simplifies the development of large-scale scientific codes, provides a rich environment for prototyping, and separates parallelism from algorithm choice. We describe the implementation of PetIGA, and exemplify its use by solving a model nonlinear problem. To illustrate the robustness and flexibility of PetIGA, we solve some challenging nonlinear partial differential equations that include problems in both solid and fluid mechanics. We show strong scaling results on up to 4096 cores, which confirm the suitability of PetIGA for large scale simulations.

Phase-field modeling of brittle and ductile fracture in shells with isogeometric NURBS-based solid-shell elements

In this paper, we investigate fracture in shells with a phase-field modeling approach. The shell model is based on solid-shell kinematics with small rotations and displacements and is discretized using quadratic Non-Uniform Rational B-Spline basis functions. Membrane and shear locking is alleviated through the Assumed Natural Strain approach. The solid-shell formulation is combined with a brittle phase-field model for elastic materials, as well as with a ductile fracture model for elasto-plastic materials exhibiting J2 plasticity with isotropic hardening. Several examples demonstrate the ability of the proposed framework to capture crack initiation, propagation, merging and branching phenomena as well as crack bulging effects in shells under different states of loading.

Isogeometric Shape Optimization of Ferromagnetic Materials in Magnetic Actuators

We present an isogeometric shape optimization method for the design of magnetic actuators. The objective is to maximize the magnetic force of actuators, which tends to be highly dependent on the geometry around air gap. The isogeometric shape optimization is known to be robust for the actuator design, since it can preserve the exact geometry during the design optimization. Using the Maxwell stress tensor method, the magnetic force is calculated through the isogeometric analysis. The analytical design sensitivity provides geometric exactness due to the use of non-uniform rational B-splines basis functions. The optimization problem is solved using a gradient-based algorithm of modified method of feasible directions. To demonstrate the robustness of the isogeometric method, the geometric dependence of magnetic forces is investigated. In addition, the isogeometric optimization method is applied to the design problems of linear and rotating actuators, which yields the optimal shape of the ferromagnetic materials around the air gap.

An extended isogeometric thin shell analysis based on Kirchhoff–Love theory

An extended isogeometric element formulation (XIGA) for analysis of through-the-thickness cracks in thin shell structures is developed. The discretization is based on Non-Uniform Rational B-Splines (NURBS). The proposed XIGA formulation can reproduce the singular field near the crack tip and the discontinuities across the crack. It is based on the Kirchhoff–Love theory where C1-continuity of the displacement field is required. This condition is satisfied by the NURBS basis functions. Hence, the formulation eliminates the need of rotational degrees of freedom or the discretization of the director field facilitating the enrichment strategy. The performance and validity of the formulation is tested by several benchmark examples.

Optimization of fiber distribution in fiber reinforced composite by using NURBS functions

This research deals with the optimization of short fibers distribution in continuum structures made of Fiber Reinforced Composite (FRC) by adopting an efficient gradient based optimization approach. Motivated by lack of non-heuristic and mesh independent optimization algorithms to obtain the optimum distribution of short fibers through a design domain, Non-Uniform Rational B-spline (NURBS) basis functions have been implemented to define continuous and smooth mesh independent fiber distribution function as well as domain discretization. Thanks to higher order (here quadratic) NURBS basis functions along with their compact support, a drastic reduction in computational time has been obtained by increasing mesh size while the accuracy of the model is maintained. Moreover combination of NURBS with sensitivity based optimization method allows a fast convergence to optimum fiber distribution layout. Minimization of elastic strain energy and maximization of fundamental frequency have been considered as objective functions for static and free vibration problems, respectively; to get the maximum fiber exploitation in the structural element. Nodal volume fraction of fiber was defined as the optimization design variable while a homogenization approach based on the random orientation of short fibers in the matrix has been adopted. Some numerical examples related to the structural response under static loading as well as the free vibration behavior are finally conducted to demonstrate the capability and reliability of the model.

Multiscale thermomechanical contact: Computational homogenization with isogeometric analysis

A computational homogenization framework is developed in the context of the thermomechanical contact of two boundary layers with microscopically rough surfaces. The major goal is to accurately capture the temperature jump across the macroscopic interface in the finite deformation regime with finite deviations from the equilibrium temperature. Motivated by the limit of scale separation, a two-phase thermomechanically decoupled methodology is introduced, wherein a purely mechanical contact problem is followed by a purely thermal one. In order to correctly take into account finite size effects that are inherent to the problem, this algorithmically consistent two-phase framework is cast within a self-consistent iterative scheme that acts as a first-order corrector. For a comparison with alternative coupled homogenization frameworks as well as for numerical validation, a mortar-based thermomechanical contact algorithm is introduced. This algorithm is uniformly applicable to all orders of isogeometric discretizations through non-uniform rational B-spline basis functions. Overall, the two-phase approach combined with the mortar contact algorithm delivers a computational framework of optimal efficiency that can accurately represent the geometry of smooth surface textures. Copyright © 2013 John Wiley & Sons, Ltd.

Isogeometric Shape Design Optimization of Geometrically Nonlinear Structures#

Using the isogeometric approach, the variational formulation of both response and sensitivity analyses for geometrically nonlinear structures is derived using the total Lagrangian formulation. The geometric properties of design are embedded in the nonuniform rational B-splines basis functions and control points whose perturbation naturally results in shape changes. Thus, exact geometric models can be utilized in both response and sensitivity analyses. The normal vector and curvature are continuous in the whole design space so that enhanced shape sensitivity can be expected. Refinements and design changes during the shape optimization are easily implemented within the isogeometric framework, which maintains exact geometry without subsequent communication with computer-aided design description. Through numerical examples, the developed isogeometric sensitivity is verified to demonstrate excellent agreement with finite difference sensitivity. Also, the proposed method works very well in various shape optimization problems.

Robust design of NLF airfoils

A robust optimization design approach of natural laminar airfoils is developed in this paper. First, the non-uniform rational B-splines (NURBS) free form deformation method based on NURBS basis function is introduced to the airfoil parameterization. Second, aerodynamic characteristics are evaluated by solving Navier–Stokes equations, and the γ-Reθt¯ transition model coupling with shear-stress transport (SST) turbulent model is introduced to simulate boundary layer transition. A numerical simulation of transition flow around NLF0416 airfoil is conducted to test the code. The comparison between numerical simulation results and wind tunnel test data approves the validity and applicability of the present transition model. Third, the optimization system is set up, which uses the separated particle swarm optimization (SPSO) as search algorithm and combines the Kriging models as surrogate model during optimization. The system is applied to carry out robust design about the uncertainty of lift coefficient and Mach number for NASA NLF-0115 airfoil. The data of optimized airfoil aerodynamic characteristics indicates that the optimized airfoil can maintain laminar flow stably in an uncertain range and has a wider range of low drag.

Free Vibration Analyses of FGM Thin Plates by Isogeometric Analysis Based on Classical Plate Theory and Physical Neutral Surface

The isogeometric analysis with nonuniform rational B-spline (NURBS) based on the classical plate theory (CPT) is developed for free vibration analyses of functionally graded material (FGM) thin plates. The objective of this work is to provide an efficient and accurate numerical simulation approach for the nonhomogeneous thin plates and shells. Higher order basis functions can be easily obtained in IGA, thus the formulation of CPT based on the IGA can be simplified. For the FGM thin plates, material property gradient in the thickness direction is unsymmetrical about the midplane, so effects of midplane displacements cannot be ignored, whereas the CPT neglects midplane displacements. To eliminate the effects of midplane displacements without introducing new unknown variables, the physical neutral surface is introduced into the CPT. The approximation of the deflection field and the geometric description are performed by using the NURBS basis functions. Compared with the first-order shear deformation theory, the present method has lower memory consumption and higher efficiency. Several numerical results show that the present method yields highly accurate solutions.

A Kriging‐based error‐reproducing and interpolating kernel method for improved mesh‐free approximations

AbstractAn error‐reproducing and interpolating kernel method (ERIKM), which is a novel and improved form of the error‐reproducing kernel method (ERKM) with the nodal interpolation property, is proposed. The ERKM is a non‐uniform rational B‐splines (NURBS)‐based mesh‐free approximation scheme recently proposed by Shaw and Roy (Comput. Mech. 2007; 40(1):127–148). The ERKM is based on an initial approximation of the target function and its derivatives by NURBS basis functions. The errors in the NURBS approximation and its derivatives are then reproduced via a family of non‐NURBS basis functions. The non‐NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation obtained in the first step. In the ERKM, the interpolating property at the boundary is achieved by repeating the knot (open knot vector). However, for most problems of practical interest, employing NURBS with open knots is not possible because of the complex geometry of the domain, and consequently ERKM shape functions turn out to be non‐interpolating. In ERIKM, the error functions are obtained through localized Kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint. Interpolating error functions so obtained are then added to the NURBS approximant. While enriching the ERKM with the interpolation property, the ERIKM naturally possesses all the desirable features of the ERKM, such as insensitivity to the support size and ability to reproduce sharp layers. The proposed ERIKM is finally applied to obtain strong and weak solutions for a class of linear and non‐linear boundary value problems of engineering interest. These illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent. Copyright © 2007 John Wiley & Sons, Ltd.

A meshless, compositional approach to shape optimal design

Meshless computational strategies for shape optimal design through the composition of behavioral fields are developed in this paper. The composition of behavioral fields is carried out in a manner that is procedurally similar to composition of primitive geometrical entities through set theoretic Boolean operations in constructive solid geometry. The methodology is shown to lead to a tighter integration between design and analysis and is termed constructive solid analysis. A meshless approximation using non-uniform rational B-spline basis functions is used to discretize the behavioral fields defined over the geometrical primitives. Remeshing, which is a critical bottleneck in shape optimization using conventional finite element analysis is either eliminated or limited to those primitives that are modified. Further, it is demonstrated that the reconstruction of stiffness matrices is required only for those geometrical primitives that are affected by the design changes. Design sensitivity analysis techniques are developed for the discretized solution system arising out of the compositional analysis. An adaptive numerical integration scheme is developed and shown to be necessary to accurately compute the effect of small domain perturbations on the problem functions. Some of the common challenges in conventional shape optimization problems such as difficulty in creating and merging multiple holes are naturally resolved using the developed compositional approach. Several representative problems are solved to validate and to demonstrate the methodology.

Analyses of wrinkled and slack membranes through an error reproducing mesh-free method

A mesh-free approximation of large deformations of flexible membrane structures within the tension field theory is considered in this paper. A modification of the wrinkling theory, originally proposed by Roddeman et al. (1987) [Roddeman, D.G., Drukker, J., Oomens, C.W.J., Janssen, J.D., 1987, The wrinkling of thin membranes: Part I—theory; Part II—numerical analysis. ASME J. Appl. Mech. 54, 884–892.], is proposed to study the behaviour of an isotropic membrane under the mixed state of stress (taut, wrinkled and slack). Using the facts that the state of stress is not uniform across an element and that the deformation gradient is a spatially continuous (and possibly non-differentiable) tensor, the proposed model uses a continuously modified deformation gradient to capture the location and orientation of wrinkles more precisely. While the deformation gradient need not be everywhere-differentiable in a wrinkled membrane, it is argued that the fictive non-wrinkled (non-slack) surface may be looked upon as an everywhere-taut surface in the limit as the minor (and major) principal tensile stresses over the wrinkled (slack) portions go to zero. Accordingly, the modified deformation gradient is thought of as the limit of a sequence of everywhere-differentiable tensors. The weighted residual from the governing equations are presently solved via a mesh-free method, where the entire domain is discretized only by a set of grid points. A non-uniform-rational-B-spline (NURBS) based error reproducing kernel method (ERKM) has been used to approximate the field variable over the domain. The first step in the method is to approximate a function and its derivatives through NURBS basis functions. However, since NURBS functions neither reproduce any polynomial nor interpolate the grid points (also referred to as control or nodal points), the approximated functions result in uncontrolled errors over the domain including the grid points. Accordingly the error functions in the NURBS approximation and its derivatives are reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation of the function obtained in the first step. Several numerical examples on wrinkled and/or slack membranes are also provided.