Non-Uniform Rational B-Spline Functions Research Articles

Accelerating tri-directional material distribution optimization in functionally graded plates with an adaptive design control point variable selection

In this paper, the problem of optimizing tri-directional material distribution in functionally graded (FG) plates under static loads to minimize the compliance is studied. Generalized shear deformation theory (GSDT) in the framework of isogeometric analysis is employed as the analyzer for efficiency and accuracy, and a non-uniform rational B-spline function is used for depicting the material distribution. As the number of control points in fine 3D design meshes, i.e., the number of optimization variables, can be too large for an algorithm to find a practically feasible design, we propose an adaptive variable selection mechanism. It gradually selects control points at important regions as variables by considering neighbor material variance, hence accelerates the optimization process. Various plate geometries are considered to validate the present approach. The findings confirm that the design mesh should be fine, and that the proposed variable selection strategy enables the optimization algorithm to find much more refined designs even in cases of large dimensions while also reducing computation.

Towards the isogeometric aero-engine

Creating very detailed finite element models of aero-engines is a very time-consuming process especially if structured meshes have to be generated for thousands of parts. Isogeometric analysis offers the possibility of an accelerated model creation process while achieving higher accuracy by using the non-uniform rational B-spline functions used for the geometry description also as basis functions for the finite element analysis. In this case, no meshing is necessary anymore. Here, we demonstrate this process by applying a self-developed tool which creates a geometry description and a computational model of a part at the same time. By the help of this tool we build up a simplified mechanical aero-engine model, which is used for transient implicit computations simulating the acceleration process of the rotor. We discuss the results as well as the computational time of isogeometric models in comparison to classical finite element models.

Application of NURBS in the Fracture Mechanics Framework to Study the Stress Intensity Factor

In this investigation, we successfully employed the Non-Uniform Rational B-spline (NURBS) to study the fracture mechanics. the NURBS functions are very popular in the design field (CAD model) and it is used as an alternative to Lagrange interpolation polynomials. The extended isogeometric analysis based on this basis function is used to evaluate the stress intensity factors (SIFs) in order to control the crack propagation. For various crack lengths, SIFs were calculated to validate the accuracy of this technique. The results obtained are in good agreement with the available techniques as CFEM, X-FEM.

NURBS-Enhanced polygonal scaled boundary finite element method for heat diffusion in anisotropic media with internal heat sources

We develop polygonal scaled boundary finite element method (SBFEM) with boundaries represented by non-uniform rational B-splines (NURBS) instead of standard Lagrange shape functions. The approach, coined NURBS enhanced SBFEM (NESBFEM), is able to represent the geometry exactly. For elements intersected by the boundary, NURBS basis functions are employed and for other elements, either Lagrange shape functions or NURBS functions are used. The proposed technique inherits the salient features of isogeometric analysis (IGA) and scaled boundary finite element method (SBFEM). Further, well-developed refinement strategies (h−,k−, p-refinements) of IGA are employed to generate higher order polytopal elements. The proposed approach provides a starting point to an efficient framework for practising engineers to solve heat diffusion in anisotropic media with internal heat sources, especially when the domain has internal features, viz., holes/inclusions and to obtain accurate solutions even on coarser mesh with high-order polytopal element.

Isogeometric indirect BEM solution based on virtual continuous sources placed directly on the boundary of 2D Helmholtz acoustic problems

An indirect boundary element method (BEM) based on isogeometric analysis (IGA) is proposed for 2D Helmholtz acoustic problems using virtual continuous sources placed directly on the problem boundary. The virtual sources form a virtual boundary identical to the main problem boundary. The proposed solution couples the CAD model with the analysis model by approximating both the CAD model and the virtual continuous sources with the same non-uniform rational B-spline functions (NURBS). Moreover, neither domain discretization nor truncation boundaries at the far-field are required. This solution creates only one coefficient matrix by directly arranging the linear system of equations.The solution follows a collocation scheme on the boundaries based on Greville abscissae with offsets wherever C0 continuity is encountered to permit an easy prediction for the normal directions and the free-terms at the collocation points required in the cases of Neumann and Robin boundary conditions. It allows us also to treat all integrals with standard Gauss quadrature points.Several numerical examples for exterior and interior acoustic problems are discussed to verify the proposed solution with comparisons to analytical solutions and previously published numerical results. These examples prove the robustness of the proposed solution even for high wavenumbers, in contrast to the previous attempts which implemented extensive investigations to find out the optimum place of the sources outside the domain producing minimum errors. Furthermore, no fictitious eigenfrequency problem is observed for exterior acoustic problems.

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.

Simulating the effect of injection well on groundwater table in unconfined aquifer using numerical model of isogeometric analysis and optimization of injection rate with PSO algorithm

Currently, one of the solutions to improve the decline in the water table of aquifers owing to their uncontrolled extraction is artificial recharge schemes, especially in arid and semi-arid regions. The present study applies a nonuniform rational B-spline functions (NURBS) based isogeometric method to investigate the effects of an injection well on the rise of water table, unconfined aquifer hydrodynamic parameters and injection rate. Also, the best injection rate was determined by linking the isogeometric analysis (IGA) simulation model to the particle swarm optimization (PSO) model. Initially, the IGA method was implemented to simulate a water table over an unconfined aquifer with two extraction wells with discharges of 1142.85 and 1428.57 m 3 day −1 for 210 days and was compared with the Modflow model. The model was tuned and run by constructing an injection well with 8214.28 m 3 day −1 injection rate for 1500 days. The IGA simulation results with ME = −0.0096, MAE = 0.0111 and RMSE = 0.0146 indicated the accuracy of the model in water table simulation compared with the Modflow model with ME = −0.0188, MAE = 0.023 and RMSE = 0.0284. The findings revealed that the use of an injection well has a positive impact on raising the water table, with a rise of 1.24 m in the observation well. Investigating the effect of the parameters on the water table changes indicated that the injection rate directly affects changes in the water table. The water table rises by more than 43% with a 50% increase in the injection rate. Also, compared with the effect of the injection rate on the water table, the transmissivity and specific yield effects were not substantial. The results of IGA–PSO showed that the best value of injection rate is 2000 m 3 day −1 , whereby the aquifer water table rises by an average of 37.12 cm.

Adaptive fully nonlinear potential model for the free surface under compressible air pressure of oscillating water column devices

A two dimensional fully nonlinear numerical wave flume is extended with a pneumatic model to compute the free surface flow inside the chamber of land-based Oscillating Water Column (OWC) devices. Potential theory and the Mixed Eulerian-Lagrangian method are employed to compute the fluid flow using the direct Boundary Element Method (BEM) enhanced by a non-uniform rational B-Spline function. Meanwhile, a pneumatic model is manipulated to obtain the instantaneous air pressure on the chamber's free surface. Spring-like behavior of a perfect gas through an adiabatic process and a linear power take-off mechanism are used in the pneumatic model that makes time derivative of the air pressure turns to a function of reciprocating air and fluid flows in the chamber. Therefore, temporary air pressure is taken into account on the fully nonlinear free surface boundary conditions implicitly along with the time marching algorithm. The hydrodynamic efficiency of an OWC interacting with second-order Stokes waves is compared with former studies to verify the present model. To assess the present model for simulating the air compressibility in a large-scale case study, the present solution is compared with experimental and mathematical results under an irregular wave. Afterwards, an OWC device is exposed to fifth-order Stokes waves in the flume to examine the numerical stability of computing the free surface flow outside and inside of the chamber.

Slow Rhythmical Brain Activity During Coordination of Cognitive Processes

A two dimensional fully nonlinear numerical wave flume is extended with a pneumatic model to compute the free surface flow inside the chamber of land-based Oscillating Water Column (OWC) devices. Potential theory and the Mixed Eulerian-Lagrangian method are employed to compute the fluid flow using the direct Boundary Element Method (BEM) enhanced by a non-uniform rational B-Spline function. Meanwhile, a pneumatic model is manipulated to obtain the instantaneous air pressure on the chamber's free surface. Spring-like behavior of a perfect gas through an adiabatic process and a linear power take-off mechanism are used in the pneumatic model that makes time derivative of the air pressure turns to a function of reciprocating air and fluid flows in the chamber. Therefore, temporary air pressure is taken into account on the fully nonlinear free surface boundary conditions implicitly along with the time marching algorithm. The hydrodynamic efficiency of an OWC interacting with second-order Stokes waves is compared with former studies to verify the present model. To assess the present model for simulating the air compressibility in a large-scale case study, the present solution is compared with experimental and mathematical results under an irregular wave. Afterwards, an OWC device is exposed to fifth-order Stokes waves in the flume to examine the numerical stability of computing the free surface flow outside and inside of the chamber.

An isogeometric analysis approach for solving the Reynolds equation in textured piston ring – cylinder liner contacts

PurposeThe purpose of this study is to propose an isogeometric analysis (IGA) approach for solving the Reynolds equation in textured piston ring cylinder liner (PRCL) contacts.Design/methodology/approachThe texture region is accurately and conveniently expressed by non-uniform rational B-splines (NURBS) besides hydrodynamic pressure and the oil film density ratio is represented in this mathematical form. A quadratic programming method combined with a Lagrange multiplier method is developed to address the cavitation issue.FindingsThe comparison with the results solved by an analytical method has verified the effectiveness of the proposed approach. In the study of the PRCL contact with two-dimensional circular dimple textures, the solution of the IGA approach shows high smoothness and accuracy, and it well satisfies the complementarity condition in the case of cavitation presence.Originality/valueThis paper proposes an IGA approach for solving the Reynolds equation in textured PRCL contacts. Its novelty is reflected in three aspects. First, NURBS functions are simultaneously used to express the solution domain, texture shape, hydrodynamic pressure and oil density ratio. Second, the streamline upwind/Petrov–Galerkin method is adopted to create a weak form for the Reynolds equation that takes the oil density ratio as a first-order unknown variable. Third, a quadratic programming approach is developed to impose the complementarity conditions between the hydrodynamic pressure and the oil density ratio.

Numerical simulation of arbitrary holes in orthotropic media by an efficient computational method based on adaptive XIGA

We present in this paper an efficient computational approach based on an adaptive extended isogeometric analysis (XIGA) for simulation of arbitrary holes in orthotropic materials. The interfaces of holes are described by multiple level set functions so that the developed XIGA can capture the hole geometry without considering its interfaces. The approach is further enhanced by using the locally refined (LR) B-spline basis functions, which dominate over B-spline or non-uniform rational B-spline functions (NURBS) due to the local refinement. This study only deals with the structured mesh strategy for local refinement. The implementation of the local refinement is guided by posteriori error estimator based on stress recovery. Accuracy study is performed for isotropic media due to the availability of analytical solutions. For numerical experiments, different types of holes in orthotropic media are studied, and the computed numerical results are compared with reference solutions derived from ABAQUS (FEM). We also compare the convergence rate obtained by our adaptive local refinement with that derived from uniform global refinement to indicate the greater advantage of the developed adaptive XIGA.

A general framework for motion design of the follower in cam mechanisms by using non-uniform rational B-spline

This paper presents a general framework for the motion design of cam mechanisms using Non-Uniform Rational B-Spline (NURBS). The follower motion is described by the NURBS function. To establish motion curves, the system of linear equations is set up by arbitrary boundary conditions of the follower motion on displacements, velocities, accelerations, and jerks. Moreover, the computation of the NURBS parameters (the knot vector and the weight factor) is formulated in order to reduce peak values of the acceleration and jerk. The example demonstrates that the motion curves used by NURBS obtain advantageous characteristics and the maximum values of the acceleration and jerk on the motion curves decrease considerably compared with the polynomial and B-spline functions. Therefore, inertial forces and the tendency of vibration can be reduced in cam dynamics of high-speed cam systems. The results show that using NURBS for synthesizing motion curves is robust and effective because this can be applied for arbitrary motion curves of cam mechanisms.

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.

Movement Epenthesis Generation Using NURBS-Based Spatial Interpolation

This study proposes a novel approach to the generation of video-based movement epenthesis for sign language and concentrates on a spatial interpolation approach using a nonuniform rational B-spline function to produce a smooth interpolation curve. To generate movement epenthesis, the beginning and end points are determined based on the concatenation cost, which is a linear combination of the distance, smoothness, and image distortion costs. The distance cost is defined as the normalized Euclidian distance between the palm locations at two cut points. The smoothness cost is determined by accumulating the second derivative of the curve. The image distortion cost is then estimated from the normalized Euclidian distance between the real and generated hand images. To evaluate the proposed approach, a set of sign video databases is collected and preprocessed with image calibration, content annotation, and principle component analysis. An image component overlapping procedure is also employed to yield a smooth sign video output. Evaluation results demonstrate that the synthesized trajectory is very close to the original trajectory. Moreover, the system was evaluated subjectively based on the results of ten hearing and five deaf people. The evaluation results also demonstrate the stability and feasibility of the proposed approach

A NURBS-based Error Reproducing Kernel Method with Applications in Solid Mechanics

A novel method for derivation of mesh-free shape functions is proposed. The first step in the method is to approximate a function and its derivatives through non-uniform-rational-B-spline (NURBS) basis functions. However since NURBS functions neither reproduce polynomials of degree higher than one nor interpolate the control points (also referred to as grid or nodal points), the approximated function leads to uncontrolled errors over the domain including the nodal points. Accordingly the error function 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. Since any desired order of continuity in the approximation can be achieved through NURBS, the proposed error reproducing kernel method (ERKM) can even approximate functions with discontinuous derivatives. Moreover, thanks to the variation diminishing property of NURBS, it has advantages in representing sharp layers without the so-called Gibbs‘ or Runge’s phenomena. Since derivatives are reproduced within polynomial spaces of appropriately reduced dimensions, differentiability requirements of the kernel functions are avoided. Any compactly supported continuous function, monotonically decreasing on either side of its maximum, may be used as the weight function (unlike other mesh free approximations). As it turns out, a target function is mainly approximated via NURBS and error functions are just supposed to add corrections, whose magnitudes are typically an order less than those of the NURBS components. The proposed method is observed to be nearly insensitive to the support size of the weight function. The proposed method is next applied to some linear and nonlinear boundary value problems of typical interest in solid mechanics. Some of these results are compared with those obtained via the standard form of RKPM. In the process, the relative numerical advantages and accuracy of the new method are brought out to an extent.