B-spline Elements Research Articles

Finite element and isogeometric stabilized methods for the advection-diffusion-reaction equation

We develop two new stabilized methods for the steady advection-diffusion-reaction equation, referred to as the Streamline GSC Method and the Directional GSC Method. Both are globally conservative and perform well in numerical studies utilizing linear, quadratic, cubic, and quartic Lagrange finite elements and maximally smooth B-spline elements. For the streamline GSC method we can prove coercivity, convergence, and optimal-order error estimates in a strong norm that are robust in the advective and reactive limits. The directional GSC method is designed to accurately resolve boundary layers for flows that impinge upon the boundary at an angle, a long-standing problem. The directional GSC method performs better than the streamline GSC method in the numerical studies, but it is not coercive. We conjecture it is inf-sup stable but we are unable to prove it at this time. However, calculations of the inf-sup constant support the conjecture. In the numerical studies, B-spline finite elements consistently perform better than Lagrange finite elements of the same order and number of unknowns.

Immersed boundary Mindlin-Reissner 3D shell element for modeling isotropic and laminated composite shells

An immersed boundary 3D shell element is presented here that is based on Mindlin-Reissner shell theory assumptions and uses quadratic B-spline approximation for the solution. It enables mesh independent analysis wherein the surface representing the shell geometry is defined independently and not by the mesh. The shell geometry is immersed in a uniform background mesh whose elements are 3D B-spline shell elements. By making the geometric model independent of the mesh, the typical difficulties associated with mesh generation can be avoided. The quadratic 3D B-spline shell elements in the mesh can represent the displacement field as a tangent continuous piece-wise polynomial approximation. The nodes of the element have three translational and no rotational degrees of freedom. The element is formulated under small-deformation and plane stress assumptions. Constructing the element based on shell theory enables the use of effective properties to model thin fiber reinforced laminated composite shell-like structures. The element formulation and the mesh independent approach are validated against a series of common shell element benchmark problems. • Immersed boundary 3D B-spline element based on Mindlin-Reissner shell theory assumptions. • Mesh independent analysis where geometry is embedded in a uniform background mesh. • The iso-parametric quadratic B-spline shell elements are used for C 1 continuous approximation. • Step boundary method (SBM) is used for applying boundary conditions. • Immersed boundary FEM for shells is validated for isotropic and composite shells.

Hydrodynamic Analysis of 3D Hydrofoil Using Nonuniform Rational B-Spline and Boundary Element Method

In this paper, two different 3D hydrofoils with profiles NACA0012 are simulated in the potential flow. Boundary element method (BEM) and nonuniform rational B-spline (NURBS) are coupled to reduce error and increase accuracy. The computer code is developed in different submergence depths (d), flow velocities (U), and various angles of attack (AoA), and the pressure is obtained by NURBS formulation. The pressure on a 3D hydrofoil with NACA412 profile iscompared with other existing methods. The validity of result is revealed. The accuracy of the results is acceptable. The competition of the two models’ results indicates that the increasing chord length leads to increase in C p min , and the decrease in depth and angle of attack leads to the growing value of C p min . Moreover, when the flow velocity is changed, the changes of potential and pressure coefficient distribution do not follow the specific trend. NURBS is a basic equation in different CAD packages because it is able to mesh surfaces. This study demonstrates that this algorithm does mesh surface of high quality, so it can be developed to generate mesh on the submerged three-dimensional bodies .

Immersed boundary thin shell analysis using 3D B-Spline background mesh

An immersed boundary shell element formulation based on Kirchhoff-Love shell theory is presented here that uses a background mesh consisting of uniform 3D B-spline elements within which the geometry and the boundaries are embedded. In this theory, rotation is assumed to be identical to slope which introduces second derivatives of deflection in the weak form. As a result, elements that provide tangent continuous approximation of the displacement are needed. A 3D shell element is formulated under small deformation assumptions that has three translational degrees of freedom (DOF) per node and uses quadratic B-Spline shape functions. The displacement field is transformed to a local coordinate system attached to the shell surface and defined such that the shell theory assumptions are satisfied. Essential boundary conditions are applied using the step boundary method where approximate step functions are utilized to construct trial and test solutions that satisfy the boundary conditions. Benchmark problems that are commonly used to test shell elements are used to validate the 3D B-spline thin shell element presented here. • 3D B-spline element formulation for modeling thin shells using an immersed boundary approach. • The shell geometry is immersed in a background mesh. • The background mesh consists of isoparametric B-spline elements. • The nodes of the element have three translational degrees of freedom and no rotations. • Step boundary method is used for applying boundary conditions.

Isogeometric MITC shell

This paper proposes an isogeometric formulation of the Reissner–Mindlin shell, using the Mixed Interpolation of Tensorial Components (MITC) technique to alleviate shear locking and membrane locking. Instead of over each Non-Uniform Rational B-Spline (NURBS) element, kinematics of the shell is directly formulated on the entire NURBS patch, with its displacements decoupled to the motions of control points and of director vectors tied to the control points. Since control points are in general not located on the surface, those control-point director vectors are interpolated from normal vectors at a set of pre-defined integration points on the patch. The assumed in-plane membrane strain field and the assumed transverse shear strain field are built as linear combinations of NURBS basis functions with degrees lower than those employed by the displacement interpolation, and tied to the original covariant strain fields at a set of well-selected tying points. Integration points for the definition of control-point director vectors and tying points of the assumed covariant strain fields are provided by the optimal and reduced macro-element quadrature rules, respectively. In this way, the classical MITC technique is fully incorporated into the isogeometric framework with arbitrary polynomial orders or patch configurations. Thanks to the high smoothness of NURBS basis functions, the geometry-preserving nature of isogeometric transformation, and the locking-resistive capability of MITC technique, the isogeometric MITC shell formulation shows superior convergence behavior, minimal geometrical error, and good robustness against patch distortion. In particular, its convergence property is insensitive to NURBS polynomial order or control point density. These advantages are demonstrated through a number of well-established benchmark problems, validating the proposed isogeometric MITC formulation being an accurate and efficient solution for linear analysis of shell structures.

Solving the Boundary Value Problem of Curves With Prescribed Geodesic Curvature Based on a Cubic B-Spline Element Method

Composite manufacturing processes such as automated tape placement (ATP) and filament winding process (FW) put forward specific requirements for the geodesic curvature along the layup paths of the composite tows and tapes. In this paper, a non-uniform cubic b-spline element method is proposed for solving the boundary value problem of curves with prescribed geodesic curvature. The differential equation system of the target curve is discretized through the point collocation method, and a quasi-Newton iteration scheme is adopted to approach the real solution from an initial approximation. The proposed method is proved to have third order accuracy, which shows more superiorities comparing with existing numerical methods. Simulations and experiments on a series of parametric surfaces are performed to investigate the performance of the proposed approach and the results verify the high efficiency. The proposed method could cope with the BVP for curves no matter their geodesic curvature vanishes or not. At the same time, the computed curves are natural and smooth such that interpolation technique is unnecessary to ensure the continuity of target curves. One potential application of this method is trajectory optimization for automated tape placement process.

Non-conforming multipatches for NURBS-based finite element analysis of higher-order phase-field models for brittle fracture

This paper proposes an effective computational tool for brittle crack propagation problems based on a combination of a higher-order phase-field model and a non-conforming mesh using a NURBS-based isogeometric approach. This combination, as demonstrated in this paper, is of great benefit in reducing the computational cost of using a local refinement mesh and a higher-order phase-field, which needs higher derivatives of basis functions. Compared with other approaches using a local refinement mesh, the Virtual Uncommon-Knot-Inserted Master-Slave (VUKIMS) method presented here is not only simple to implement but can also reduce the variable numbers. VUKIMS is an outstanding choice in order to establish a local refinement mesh, i.e. a non-conforming mesh, in a multi-patch problem. A phase-field model is an efficient approach for various complicated crack patterns, including those with or without an initial crack path, curved cracks, crack coalescence, and crack propagation through holes. The paper demonstrates that cubic NURBS elements are ideal for balancing the computational cost and the accuracy because they can produce accurate solutions by utilising a lower degree of freedom number than an extremely fine mesh of first-order B-spline elements.

Applying isogeometric approach for free vibration, mechanical, and thermal buckling analyses of functionally graded variable-thickness blades

This article presents free vibration and buckling analyses of functionally graded blades with variable thickness subjected to mechanical and thermal loading using isogeometric analysis as a powerful numerical method. The proposed method is based on deployment of Hamilton’s principle to the two-dimensional kinematics of blades. The governing equations are derived in the context of a modified form of higher order shear deformation plate theory that merely needs C0-continuity (C0-higher order shear deformation plate theory). Without the necessity of defining a shear correction factor, the theory can accurately predict the solution for different thickness-to-length ratios. The numerical predictions for the buckling loads and natural frequencies are successfully compared with the available solutions in the published articles and in the lack of relevant results, finite element analysis using ANSYS is used for verification of the model. The effects of variable thickness and aspect ratio on the natural frequencies and mode shapes known as the frequencies loci veering phenomena are assessed for the first time, which is an important design factor for the blades. The proposed method uses non-uniform rational B-spline element, which is able to approximate linear and nonlinear thickness distribution and the couple modes with an excellent numerical consistency. The influences of aspect ratio, thickness variation, taper ratio, volume fraction exponent, and boundary conditions on the free vibration and buckling of variable-thickness functionally graded blades are also examined.

Isoparametric B-Spline Elements for Immersed Boundary Explicit Dynamic Simulation

AbstractExplicit dynamic analysis has proven to be advantageous when simulating shock and impact loading, and very small time-scale events. In this article, the feasibility of using a background mesh of B-spline elements for immersed boundary explicit dynamic simulation is studied. In this approach, the geometry is immersed in a background mesh consisting of uniform regular shaped elements to avoid mesh generation difficulties. The boundary conditions are applied using the step boundary method, which uses the equations of the boundaries to construct trial functions that satisfy the essential boundary conditions. An isoparametric formulation is presented for quadratic and cubic B-spline elements and their shape functions are derived from the classical recursive definition of B-splines. The effectiveness of mass diagonalization for B-spline elements is also explored. This approach is validated using several examples by comparing with modal superposition solutions as well as past work using traditional finite element analysis (FEA) and analytical solutions when available.

Survey of Immersed Boundary Approaches for Finite Element Analysis

AbstractMesh generation for traditional finite element analysis has proven to be very difficult to fully automate especially using hexahedral elements for complex 3D geometry. Several modifications to the finite element method (FEM), such as the meshless methods, have been proposed for avoiding mesh generation. An alternative approach has recently gained popularity where the geometry, created as a solid model in cad software, is embedded or immersed in a nonconforming background mesh for analysis. In this approach, referred to here as the immersed boundary approach, a background mesh that is independent of the geometry is used for piecewise interpolation or approximation of the solution. Therefore, a uniform mesh with regular-shaped or undistorted elements can be used, and such a mesh is easy to generate automatically. When the geometry is immersed in the background mesh, the boundary elements are often only partly inside the geometry and the nodes of the mesh may not be on the boundaries. Many new methods have been developed to integrate over partial elements and to apply boundary and interface conditions when the boundaries of the geometries do not conform to the background mesh. These methods are reviewed in this article with particular emphasis on the implicit boundary method and step boundary method for applying boundary conditions. In addition, B-spline elements and several applications of the immersed boundary approach are surveyed including composite microstructures and structural elements for plates and shells.

A Computational Framework for Design and Optimization of Flexoelectric Materials

We combine isogeometric analysis (IGA), level set (LS) and pointwise density-mapping techniques for design and topology optimization of piezoelectric/flexoelectric materials. We use B-spline elements to discretize the fourth-order partial differential equations of flexoelectricity, which require at least [Formula: see text] continuous approximations. We adopt the multiphase vector LS model which easily copes with various numbers of material phases and multiple constraints. In case studies, we first confirm the accuracy of the IGA model and then provide numerical examples for both pure and composite flexoelectric structures. The results demonstrate the significant enhancement in electromechanical coupling coefficient that can be obtained using topology optimization and particularly by multi-material topology optimization for flexoelectric composites.

LR B-Splines implementation in the Altair RadiossTM solver for explicit dynamics IsoGeometric Analysis

IsoGeometric Analysis has shown to be a very promising tool for an integrated design and analysis process. A challenging task is still to move IGA from a proof of concept to a convenient design tool for industry and this work contributes to this endeavor. This paper deals with the implementation of the IGA concept into Altair Radioss finite element solver in order to address crash and stamping simulation applications. To this end, the necessary ingredients to a smooth integration of IGA in a traditional finite element code have been identified and adapted to the existing code architecture. A solid B-Spline element has been developed in Altair Radioss and then, an existing contact interface has been extended in order to work seamlessly with both NURBS and Lagrange finite elements. As local refinement is needed for solution approximation, an analysis is made in terms of analysis suitability and implementation aspects for several Spline technologies. The Locally Refined B-Spline (LRBS) technology is implemented and is validated on industrial benchmarks, for validation cases conventionally used for industrial codes like stamping and drop test.

3D thermomechanical buckling analysis of perforated annular sector plates with multiaxial material heterogeneities based on curved B-spline elements

In the present paper, thermomechanical buckling of perforated functionally graded annular sector plates under uniform temperature rises and radial, circumferential or biaxial mechanical loads are investigated. Furthermore, effects of the circular cutouts on the buckling strength and deformation pattern are studied. Influence of the heterogeneity of material of the FGM sector in radial, circumferential, and transverse directions on the thermomechanical buckling is investigated for plates without or with one or two holes. Governing equations are derived based on the 3D energy-based theory of elasticity and buckling occurrence is detected by Treftz instability criterion. A novel curved 3D B-splined C2-continuous element is proposed to trace the inter-element variations of the displacements and stresses and model the geometric discontinuities (cutouts). A proper algorithm is also proposed to relate events of the original cylindrical coordinates to those of the natural coordinates and vice versa. Instead of using the common von Karman assumptions, the most general form of the strain tensor in curvilinear coordinates is used. Finally, effects of the sector dimensions, size and orientation of the cutout, heterogeneity direction, direction of the mechanical loads and the combination of the thermal and mechanical loads on the buckling loads and mode shapes are investigated.

A multi-material level set-based topology optimization of flexoelectric composites

We present a computational design methodology for topology optimization of multi-material-based flexoelectric composites. The methodology extends our recently proposed design methodology for a single flexoelectric material. We adopt the multi-phase vector level set (LS) model which easily copes with various numbers of phases, efficiently satisfies multiple constraints and intrinsically avoids overlap or vacuum among different phases. We extend the point wise density mapping technique for multi-material design and use the B-spline elements to discretize the partial differential equations (PDEs) of flexoelectricity. The dependence of the objective function on the design variables is incorporated using the adjoint technique. The obtained design sensitivities are used in the Hamilton–Jacobi (H–J) equation to update the LS function. We provide numerical examples for two, three and four phase flexoelectric composites to demonstrate the flexibility of the model as well as the significant enhancement in electromechanical coupling coefficient that can be obtained using multi-material topology optimization for flexoelectric composites.

Isogeometric Analysis of the Navier–Stokes equations with Taylor–Hood B-spline elements

This paper presents our numerical results of the application of Isogeometric Analysis (IGA) to the velocity–pressure formulation of the steady state as well as to the unsteady incompressible Navier–Stokes equations. For the approximation of the velocity and pressure fields, LBB compatible B-spline spaces are used which can be regarded as smooth generalizations of Taylor–Hood pairs of finite element spaces. The single-step θ-scheme is used for the discretization in time. The lid-driven cavity flow, in addition to its regularized version and flow around cylinder, are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme.

Hybrid volume completion with higher-order Bézier elements

This paper proposes a methodology to create a hybrid volumetric representation from a 2-manifold without boundaries represented with untrimmed B-spline surfaces. The product consists of trivariate tensor product B-splines near the boundary and unstructured higher-order Bézier tetrahedral elements in the interior of the object with C0 smoothness across their interfaces. The B-spline elements are constructed by offsetting the input surface into its interior. Then, an intermediate interface consisting of Bézier triangles is built to match the inner boundary of the B-spline representation. The rest of the space, bounded by the intermediate interface, is then filled with unstructured Bézier tetrahedral elements. Our approach to constructing stiffness and mass matrices takes into account the dependencies of interior Bézier tetrahedral elements on exterior B-spline elements along their interface, so that C0 smoothness is automatically maintained when performing computer graphics simulations, representing material properties, or performing isogeometric analysis. We apply our methodology on a variety of 3-D objects and demonstrate a fast convergence rate with our 2-D studies.

3D B-spline finite element nonlinear elasticity buckling analysis of rectangular FGM plates under non-uniform edge loads, using a micromechanical model

Three-dimensional buckling analysis of the functionally graded plates has not been investigated under non-uniform in-plane compressive loads so far. The only available work in this field has been performed based on the first-order shear deformation theory, employing the simple rule of mixtures. In the present paper, a non-linear three-dimensional energy-based elasticity analysis is developed for buckling investigation of functionally graded plates subjected to non-uniform in-plane compressions. The comparative studies performed in the present paper reveal that the rule of mixtures is generally inappropriate for determination of the effective material properties. For this reason, a micromechanics-based model is used instead. In contrast to the common displacement-based numerical or semi-analytical elasticity analyses (with C0-continuity), present formulations are C2-continuous due to using the proposed 3D cubic B-spline element. Buckling analysis is accomplished through a two-step procedure wherein the prebuckling stresses are determined first. A non-linear weighted-residuals-based finite element solution is used. The buckling load associated with each of the adopted 8 types of the load distribution patterns is detected based on a generalized geometric stiffness concept. Various and comprehensive parametric studies are accomplished and various boundary conditions are considered to extract more precise conclusions.

Sliding and Nonsliding Joint Constraints of B-Spline Plate Elements for Integration With Flexible Multibody Dynamics Simulation

In this investigation, a numerical procedure for modeling sliding and nonsliding joint constraints for the B-spline thin plate element is developed for the large deformation analysis of multibody systems. A concept of intermediate reference coordinates proposed for the absolute nodal coordinate formulation is generalized for B-spline elements such that a wide variety of joint constraints can be modeled using existing joint constraint libraries already implemented in multibody dynamics codes. This procedure allows for modeling sliding joints for B-spline elements that requires a solution to moving boundary problems by introducing time-variant surface parameters in the B-spline parametric domain. Since surface parameters treated as knot variables in the basis function are defined in the entire parametric domain rather than the element domain, the location of the constraint definition point can be determined without knowing in which elements the sliding point is located. Furthermore, using the B-spline recurrence formula, control points used for describing the constraint equations can be systematically extracted. It is shown that many types of nonsliding joints fixed on the flexible body can also be modeled as a special case of the sliding joint formulation developed in this investigation, leading to a unified joint constraint formulation for B-spline elements. Several numerical examples are presented in order to demonstrate the use of the numerical procedure developed in this investigation.

Three-dimensional non-linear elasticity-based 3D cubic B-spline finite element shear buckling analysis of rectangular orthotropic FGM plates surrounded by elastic foundations

In the present paper, shear buckling analysis of the orthotropic heterogeneous FGM plates is investigated for the first time. Moreover, influence of the Winkler-type elastic foundation is considered. The material properties are assumed to have in-plane orthotropy and transverse heterogeneity. The most accurate approach, i.e., the three-dimensional elasticity is employed instead of using the approximate plate theories. In contrast to all of the available displacement-based buckling analyses that have employed C0-continuous commercial finite element codes or semi-analytical methods, present formulations are C2-continuous due to using the proposed 3D cubic B-spline element. Results are derived based on principle of minimum potential energy and a non-linear finite element procedure utilizing a Galerkin-type 3D cubic B-spline solution algorithm. Buckling loads are detected based on a generalized geometric stiffness concept. In this regard, effects of both the prebuckling and buckling states are considered. To present a better imagination and more detailed discussions, details of the buckling mode shapes and the foundation interaction are discussed for plates with simply-supported edges. In addition, the more practical free and clamped edge conditions are also considered.

Reproducing kernel triangular B-spline-based FEM for solving PDEs

We propose a reproducing kernel triangular B-spline-based finite element method (FEM) as an improvement to the conventional triangular B-spline element for solving partial differential equations (PDEs). In the latter, unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the B-spline. The performance therefore becomes unstable and cannot be controlled in a desirable way. To address this issue, the proposed improvement adopts the reproducing kernel approximation in the calculation of B-spline kernel function. Three types of PDE problems are tested to validate the present element and compare against the conventional triangular B-spline. It has been shown that the improved triangular B-spline satisfies the partition of unity condition even for extreme conditions including corners and holes.