Non-uniform Rational B-spline Basis Functions Research Articles

Isogeometric boundary element method for steady-state heat transfer with concentrated/surface heat sources

An isogeometric boundary element method (IGABEM) is proposed for solving the steady-state heat transfer problems with concentrated/surface internal heat sources. The isogeometric boundary element method (IGABEM) possesses the advantages of both the isogeometric analysis (IGA) and the boundary element method (BEM), the non-uniform rational B-spline (NURBS) basis functions used in the (computer-aided design) CAD system are flexible and stable in dealing with irregular boundaries, due to which the NURBS basis functions are applied to exactly reconstruct the boundary geometry of the analysis domain, and the physical variables are also approximated by the NURBS as the standard IGA does. Dirac delta function is introduced in the present IGABEM for dealing with the concentrated point heat sources, and a local coordinate system is proposed for the domain integration over line heat source based on the NURBS basis functions. As to the domain integrals caused by the surface heat sources, the radial integration method (RIM) is used to transform the domain integrals to boundary ones without any internal cells for the original version of IGABEM. Several numerical examples involving Dirichlet, Neumann and Robin boundary conditions are analyzed, and circular region with point/line heat source(sources), global and local surface heat sources within multiply connected regions are also considered. Verification of the proposed method has been gained by comparing the present results with those obtained by several other researchers, and solutions achieved by analytical expression together with the fine-meshed ANSYS model are also utilized for the purpose of comparison. Meanwhile, the computational efficiency and convergence of the present method is evaluated by comparing the IGABEM with the finite element method (FEM) as well as the traditional BEM. Good performance of the IGABEM is observed. Finally, heat transfer within a simplified printed circuit board (PCB) with both concentrated (line and point) and surface heat generations is studied, the temperature distribution is achieved, which verifies the applicability of the proposed technoque in practical engineering.

Phase-field model of brittle fracture in Reissner–Mindlin plates and shells

In this paper, a phase-field model is presented for the description of brittle fracture in a Reissner–Mindlin plate and shell formulation. The shell kinematics as well as the phase-field variable are described on the midsurface of the structure. Non-Uniform Rational B-Spline basis functions are used for the discretization of both the displacement/rotations and the phase-field. The spectral decomposition for the tension–compression split is applied on the total strain tensor, which varies through the thickness. Thus, the plane stress condition has to be enforced numerically. Various numerical examples are presented in order to verify the accuracy and effectiveness of the method and a detailed comparison to existing formulations is performed.

On Investigation of Liquid Sloshing in Cylindrical Tanks With Single and Multiply Connected Domains Using Isogeometric Boundary Element Method

Abstract This paper explores an isogeometric boundary element method (IGA-BEM) for sloshing problems in cylindrical tanks with single and multiply connected domains. Instead of the Lagrange basis functions used in the standard BEM, the nonuniform rational B-splines (NURBS) basis functions are introduced to approximate the geometries of the problem boundaries and the unknown variables. Compared with the Lagrange basis functions, NURBS basis functions can accurately reconstruct the geometric boundary of analysis domain with almost no error, and all the data information for NURBS basis functions can be directly obtained from the computer-aided design (cad) or computer-aided engineering (cae) commercial software, which implies the modeling process of IGA-BEM is more simple than that of the standard BEM. NURBS makes it possible for the IGA-BEM to realize the seamless connection between cad and cae software with relative higher calculation accuracy than the standard BEM. Based on the weighted residual method as well as the divergence theorem, the IGA-BEM is developed for the single and multiply connected domains, whose boundaries are separately defined in the parameter space by different knot vectors. The natural sloshing frequencies of the liquid sloshing in a circular cylindrical tank with a coaxial or an off-center circular pipe, an elliptical cylindrical tank with an elliptical pipe, a circular cylindrical tank with multiple pipes are estimated with the introduced method by assuming an ideal (inviscid and incompressible) liquid, irrotational small-amplitude sloshing, and the linear free-surface condition. The comparison between the results obtained by the proposed method and those in the existing literatures shows very good agreements, which verifies the proposed model well. Meanwhile, the effects of radius ratio, liquid depth, number, and location of internal pipe (pipes) on the natural sloshing frequency and sloshing mode are analyzed carefully, and some conclusions are outlined finally.

Isogeometric boundary element analysis and shape optimization by PSO for 3D axi-symmetric high frequency Helmholtz acoustic problems

The boundary element method (BEM) has been widely used for approximating the solution of the Helmholtz acoustic equation due to its superiority for handling unbounded domains by modeling only the inner boundaries and avoiding the truncation error, present in the finite element method (FEM). Non-uniform rational B-splines basis functions (NURBS) are used in an isogeometric setting to describe both the CAD geometries and the physical fields. Isogeometric analysis (IGA) is coupled with BEM forming (IGABEM) which speeds up meshing and gives higher accuracy in comparison with standard BEM. In this work, we consider axi-symmetric domains, where the three-dimensional problem is simplified in BEM from a surface integral to a combination of two 1D integrals. The first is the line integral similar to a two-dimensional BEM problem. The second integral is performed over the angle of revolution. The discretization is applied only to the former integration. This leads to significant computational savings and, consequently, better treatment for higher frequencies over the full three-dimensional models. We first validate the capability of IGABEM for axi-symmetric problems to handle high frequencies and verify it by means of manufactured analytical solutions. Secondly, we show the performance of the proposed approach in shape optimization problems by using a particle swarm optimization (PSO) method. Coupling PSO with IGABEM in the optimization models benefits from the IGA feature of representing the three different models: shape design, analysis and optimization models using a single set of control points. The proposed method for the shape optimization of a horn is compared with reference solutions for BEM and FEM.

Shape optimization by conventional and extended isogeometric boundary element method with PSO for two-dimensional Helmholtz acoustic problems

In this paper, a new approach is developed for applications of shape optimization on the two-dimensional time harmonic wave propagation (Helmholtz equation) in acoustic problems. The particle swarm optimization (PSO) algorithm - a gradient-free optimization method avoiding the sensitivity analysis - is coupled with two boundary element methods (BEM) and isogeometric analysis (IGA). The first method is the conventional isogeometric boundary element method (IGABEM). The second method is the eXtended IGABEM (XIBEM) enriched with the partition-of-unity expansion using a set of plane waves. In both methods, the computational domain is parameterized and the unknown solution is approximated using non-uniform rational B-splines basis functions (NURBS).In the optimization models, the advantage of IGA is the feature of representing the three models; i.e. shape design/analysis/optimization, using a set of control points, which also represent control variables and optimization parameters, making communication between the three models easy and straightforward.A numerical example is considered for the duct problem to validate the presented techniques against the analytical solution. Furthermore, two different applications for various frequencies are studied; the vertical noise barrier and the horn problems, and the obtained results are compared against previously published numerical methods using sensitivity analysis and genetic algorithms to verify the efficiency of the proposed approaches.

A surface-to-surface contact search method enhanced by deep learning

This paper describes a new surface-to-surface contact search method for contact problems between curved surfaces defined by the non-uniform rational B-spline basis functions. The method consists of a global search method based on the hierarchical virtual axis aligned bounding boxes and a local contact search method enhanced by the deep learning. The artificial neural network is employed to simulate the mapping from the parameters of contacting surface to the contact state: whether contact occurs or not, where the contact occurs and how far the slave surface penetrates to the master surface. With the proposed method, the time-consuming local contact search is successfully performed independently from the dynamic contact analysis part, where its computational efficiency is quantitatively discussed through some numerical examples.

Maximum energy dissipation for elasto-plastic plates via isogeometric shape optimization

In this research, optimum shape of plate structures is sought to maximize the energy dissipation via structural shape optimization. To achieve this, isogeometric analysis (IGA) is utilized for structural analysis of plates considering elasto-plastic behavior of materials. The von Mises material model is employed for this purpose. Non-uniform rational B-splines basis functions are used for both geometry definition and approximating the unknown deformation field. The optimization problem is to maximize the structural dissipated energy until a prescribed displacement is reached and a fixed amount of material is considered in the design domain. A direct shape sensitivity analysis is performed and a mathematical based approach is employed for the optimization process. To demonstrate the efficiency of the proposed algorithm three examples are illustrated. Using the IGA prevents adjusting analysis model during the optimization process, which is time-consuming especially when iterative nonlinear analysis is performed. The results also show that large geometry modifications can be properly managed by the proposed algorithm.

Isogeometric analysis of laminated smart shell structures covered with piezoelectric sensors and actuators using degenerated shell formulation

An isogeometric approach to the analysis of laminated composite smart shell structures based on the degenerated formulation and Mindlin–Reissner assumptions using non-uniform rational B-spline basis functions is the subject of this article. To model the laminated orthotropic smart free-form shells, the equivalent single layer theory is adopted, and an accurate approach to construct the local basis systems is used. To consider the electric potential in the piezoelectric layers, a sub-layer approach is employed that assumes linear variation over the thickness of the sub-layer. To investigate the performance of the approach, static, free vibration, and static control analysis of laminated composite shells covered with piezoelectric sensor and actuator layers with different degrees of basis functions is performed. Also, the effect of mechanical loading, various input voltages, and different boundary conditions on the static response and natural frequencies have been investigated. Several numerical examples are presented to demonstrate the efficiency and accuracy of the approach and validated with the existing results from the literature.

Isogeometric boundary element method for crack propagation based on Bézier extraction of NURBS

Isogeometric boundary element method (IGABEM) based on Bézier extraction of non-uniform rational B-splines (NURBS) is developed to analyze the crack propagation and the interaction between inclusions and cracks in 2D infinite isotropic medium. We use isogeometric analysis (IGA) with NURBS to describe the geometric boundaries of inclusions and cracks which can be represented exactly no matter how coarse the discretization of the studied bodies is, so there are no geometric errors. Meanwhile, the same NURBS basis functions are used to approximate the unknown boundary quantities. For the crack boundary, the collocation points are moved inside the crack elements to meet the continuity requirement, i.e., discontinuous quadratic elements are applied on the crack boundary. Then, in order to obtain accurate stress intensity factors (SIFs), special crack tip basis functions are adopted to simulate the r variation of displacements near the crack tips. Some numerical examples are given and the obtained results have been compared with the results in the references.

Hybrid Isogeometric-Finite Element Discretization Applied to Stress Concentration Problems

Isogeometric analysis (IGA) employs non-uniform rational B-splines (NURBS) or other B-spline-based variants to represent both the geometry and the field variable. Exact geometry representation and higher order global continuity (at least [Formula: see text] even on elements’ boundaries) are two favorable properties that would make IGA an appropriate discretization technique in problems with responses associated with the derivatives of the primary field variable. As a category of these problems, in this paper, 2D elastostatic problems involving stress concentration sites are analyzed with a hybrid isogeometric-finite element (IG-FE) discretization. To exploit higher order continuity of NURBS basis functions, IGA discretization is applied selectively at pre-identified locations of high displacement gradients where the stress concentration occurs. In addition, considering computational efficiency, the rest of problem domain is discretized by means of linear Lagrangian finite elements. The connection of NURBS and Lagrangian domain is carried out through employing specially devised elements [Corbett, C. J. and Sauer, R. A. [2014] “NURBS-enriched contact finite elements”, Computer Methods in Applied Mechanics and Engineering 275, 55–75]. The methodology is applied in some 2D elastostatic examples. Increasing the number of DOFs and comparing convergence of the concentrated stress value using different discretizations, it is shown that the hybrid IG-FE discretizations generally have faster and more stable convergence response compared with pure FE discretizations especially at lower DOFs.

Fatigue life estimation and crack propagation analysis of orthotropic lamina using XIGA methodology

ABSTRACTIn the present work, a NURBS-based Extended Isogeometric Analysis (XIGA) code has been developed for fracture analysis of orthotropic lamina. The XIGA code is capable of an efficient analysis of crack problems using non-uniform rational B-spline basis functions for both the solution field and geometric description of model. The level set approach is used to track the crack geometry. Crack faces are enriched by Heaviside enrichment function and crack tip singularity is captured by asymptotic crack tip enrichment functions. The interaction integral (M-Integral) method is used to calculate mixed mode stress intensity factor for further stable crack growth. The Paris crack growth law is used to simulate the crack propagation and fatigue life estimation in orthotropic media using life cycle increment approach. The developed code has been validated for static and fatigue analysis. The effect of varying angle of orthotropy on fatigue life of 2D plate is investigated using XIGA methodology leading to an optimized ply orientation of orthotropic plate.

A novel interface integral formulation for 3D steady state thermal conduction problem for a medium with non-homogenous inclusions

A novel regularized interface integral equation for three-dimensional steady state heat conduction problems with non-homogeneous inclusions is developed. The proposed formulation only contains the fundamental solution of isotropic matrix. As a result, the fundamental solution of non-homogeneous inclusion, usually very difficult to obtain, is avoided. Domain integrals caused by the contrast of heat conductivities between the inclusions and the matrix are converted into equivalent interface integrals using the radial integration method by expressing the temperature gradient as a series of radial basis functions. Therefore, a pure interface integral equation is obtained and there is no need to discretize the inclusion into finite elements to evaluate the domain integral. For the determination of the flux and temperature, collocation points are distributed inside the inclusion to form a system of linear equations. To eliminate the geometrical errors and study the inclusions with arbitrary geometry, bivariate Non-Uniform Rational B-Splines basis functions are used to depict the boundaries of the inclusions. Numerical results are compared with available analytical solutions or finite element solutions.

Analysis of cracked plate using higher-order shear deformation theory: Asymptotic crack-tip fields and XIGA implementation

In this work, an extended isogeometric analysis (XIGA) is used for the analysis of through-thickness crack in a homogeneous and isotropic plate. In isogeometric analysis (IGA), non-uniform rational B-splines (NURBS) are used as a basis function. The plate kinematics is modelled by Reddy’s higher-order shear deformation theory (HSDT). The C1 continuity requirement of HSDT can be easily fulfilled by the NURBS basis functions. In order to obtain the plate fracture parameters (moment intensity factors), the expressions of crack-tip fields (auxiliary fields) are derived using separation of variables and Eigen-function approach. A new expression for moment intensity factors is developed using auxiliary fields solution (crack-tip fields) and interaction integral approach. Several cracked plate problems are solved by XIGA using HSDT. The results obtained by HSDT based XIGA (HSDT-XIGA) are compared with the FSDT based XIGA (FSDT-XIGA) and literature solutions.

NURBS-augmented finite element method for stability analysis of arbitrary thin plates

In the analysis of a plate, the geometry plays a very important role. The non-uniform rational B-spline (NURBS) basis functions are employed for the representation of the geometry and field variables in the isogeometric analysis. These basis functions are able to represent the geometry accurately. They are non-interpolating in nature, and hence do not satisfy the Kronecker-Delta property. Hence, it becomes difficult to enforce the essential boundary conditions at the control variables. A new method called NURBS-augmented finite element method (NAFEM) was proposed (Mishra and Barik, Comput Struct, https://doi.org/10.1016/j.compstruc.2017.10.011 , 2017) and arbitrary shaped plates were successfully dealt for bending analysis. In the NAFEM, the authors adopted the finite element basis functions for the field variables as they satisfy the Kronecker-Delta property so that the boundary conditions were enforced with ease and the NURBS basis functions were employed for the geometry, thereby representing the shape of the plate accurately. In the present work, the same is extended for stability analysis of plates having different geometries and boundary conditions and the results are found to be in excellent agreement with the existing ones. Some new shapes have also been considered, and the new results are presented.

NE-IIBEFM for problems with body forces: A seamless integration of the boundary type meshfree method and the NURBS boundary in CAD

A NURBS-enhanced interpolating boundary element-free method (NE-IIBEFM) is proposed by coupling the meshfree method with the non-uniform rational B-spline (NURBS). The NURBS plays an essential role in computer aided design (CAD) whereby it can be used as the bridge between boundary type meshless method and CAD, which is convenient and is widely utilized in engineering. The NURBS basis functions used in CAD packages for geometry construction are applied to describe the boundary in the numerical procedure. Thus, the geometry of the curve can be reproduced exactly at all stages. The boundary integral cells and nodes are defined from the knot vector in NURBS, and both the open and closed curves are considered. The fields are approximated by an improved interpolating moving least square method, in which the obtained shape functions have the property of delta function. Thus, the boundary conditions can be imposed directly. A line integration method without domain discretization is further coupled with the proposed method for the treatment of domain integrals for problems with body forces.

Investigations on crack tip plastic zones by the extended iso-geometric analysis

This paper investigates the effect of holes on the shape and size of crack tip plastic zones in the engineering components being subjected to static loads by using the extended iso-geometric analysis (XIGA). The geometry and solutions are approximated by using the non-uniform rational B-spline basis functions. The conventional iso-geometric approximations are modified by adding appropriate enrichment functions, which depend upon the nature of the discontinuities present in the domain. In this method, the modelling of discontinuities created by holes and cracks becomes independent of the grid chosen for analysis. The domain based interaction integral has been used to evaluate the stress intensity factors under mixed mode loadings. The crack tip plastic zones have been obtained by employing the von-Mises and Tresca yield criteria. Finally, several numerical problems have been solved by the extended iso-geometric technique to investigate the effect of holes on the extent of crack tip plastic zones

Free vibration and buckling analysis of composite laminated plates using layerwise models based on isogeometric approach and Carrera unified formulation

ABSTRACTIn this paper, layerwise (LW) theory has been utilized along with that of equivalent single layer (ESL) for free vibration and linearized buckling analysis of composite laminated plates. To this end, the isogeometric approach and Carrera unified formulation (CUF) have been combined. In particular, Taylor-like and Legendre-like polynomial expansions have been utilized in the framework of CUF to approximate the solution field in ESL and LW models, respectively. Indeed, CUF provides an ideal tool that facilitates the implementation of higher orders of the solution field expansion. As ESL model cannot inherently provide interlaminar continuity, they are not suitable for analyzing thick laminated plates. However, the LW model not only presents a three-dimensional (3D)-like accurate mathematical model using two-dimensionalplate theories but also considers interlaminar continuity requirements and obviates the need for the use of shear correction factor. In addition, the nonuniform rational B-spline basis functions have been employed to approximate the solution field, due to their interesting attributes in the analysis. These functions are able to describe the exact geometry of the structure and make it technically feasible to provide refinement process during analysis. The presented numerical results confirm the validity of the proposed methodology.

Transient heat conduction analysis using the NURBS-enhanced scaled boundary finite element method and modified precise integration method

The Non-uniform rational B-spline (NURBS) enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this paper. The scaled boundary finite element method is a semi-analytical technique, which weakens the governing differential equations along the circumferential direction and solves those analytically in the radial direction. In this method, only the boundary is discretized in the finite element sense leading to a reduction of the spatial dimension by one with no fundamental solution required. Nevertheless, in case of the complex geometry, a huge number of elements are generally required to properly approximate the exact shape of the domain and distorted meshes are often unavoidable in the conventional finite element approach, which leads to huge computational efforts and loss of accuracy. NURBS are the most popular mathematical tool in CAD industry due to its flexibility to fit any free-form shape. In the proposed methodology, the arbitrary curved boundary of problem domain is exactly represented with NURBS basis functions, while the straight part of the boundary is discretized by the conventional Lagrange shape functions. Both the concepts of isogeometric analysis and scaled boundary finite element method are combined to form the governing equations of transient heat conduction analysis and the solution is obtained using the modified precise integration method. The stiffness matrix is obtained from a standard quadratic eigenvalue problem and the mass matrix is determined from the low-frequency expansion. Finally the governing equations become a system of first-order ordinary differential equations and the time domain response is solved numerically by the modified precise integration method. The accuracy and stability of the proposed method to deal with the transient heat conduction problems are demonstrated by numerical examples.

NURBS-enhanced boundary element method based on independent geometry and field approximation for 2D potential problems

Non-uniform rational B-spline (NURBS) in Isogeometric analysis (IGA) is coupled with the boundary element method (BEM) for 2D potential problems in this paper. The geometry and field are usually approximated by the same basis functions in IGA, such as the B-spline or the NURBS basis functions. In the proposed method, these two kinds of approximation are performed independently, i.e. the geometry is reproduced by the NURBS basis functions while the field is approximated by the traditional Lagrangian basis functions which are used in the conventional BEM. The proposed method has the advantage that the geometry can be reproduced exactly at all stages in IGA methods. Actually, one can use the computer aided design (CAD) software or NURBS library to perform the operations related to the geometry. The field approximation is performed in parameter space and separated from the geometry. Thus, it can be implemented easily as the conventional BEM since most algorithms for BEM can be applied directly, such as the methods for treatment of the singular integrals, and the boundary conditions can be imposed directly. Numerical examples have demonstrated the accuracy of the proposed method.

A NURBS-based scaled boundary finite element method for the analysis of heat conduction problems with heat fluxes and temperatures on side-faces

The scaled boundary finite element method (SBFEM) combined with isogeometric analysis (IGA) is proposed to solve the two-dimensional steady-state heat conduction problems in complex geometries. The main benefit of SBFEM is that the spatial dimension of analyzed domain is reduced by one and the solution is analytical in the radial direction. In this method, only the boundary of the computational domain requires discretization with finite elements leading to the reduction of computational efforts. However, SBFEM suffers from the finite element method related drawbacks. In the case of the complex geometric shapes, a large number of elements are necessary to obtain the exact representation of geometry in finite element method. Isogeometric analysis is a novel numerical technique based on the non-uniform rational B-splines (NURBS), where the geometry can be exactly represented. Moreover, this technique yields superior numerical accuracy, efficiency and convergence property in comparison to finite element method. In the proposed method, the segments of domain boundary with complex geometries are described with NURBS basis functions in IGA, while the straight segments of boundary are represented with polynomial basis functions as in the conventional SBFEM. Thus, the present approach combines the advantages of both SBFEM and IGA. The heat conduction problems of complex geometry can be more effectively handled with the proposed method considering the prescribed heat fluxes and temperatures on side-faces. The accuracy and efficiency of the proposed formulation are demonstrated by modeling five numerical examples involving the complicated geometry.