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Keywords:

  • shells;
  • solid-like shell element;
  • isogeometric analysis;
  • Bézier extraction;
  • geometrically nonlinear analysis

SUMMARY

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 SOLID-LIKE SHELL FORMULATION
  5. 3 ISOGEOMETRIC FINITE ELEMENT DISCRETIZATION
  6. 4 NUMERICAL SIMULATIONS
  7. 5 CONCLUDING REMARKS
  8. APPENDIX A: STRAIN COMPONENTS
  9. APPENDIX B: VIRTUAL STRAINS AND DERIVATIVES
  10. REFERENCES

An isogeometric solid-like shell formulation is proposed in which B-spline basis functions are used to construct the mid-surface of the shell. In combination with a linear Lagrange shape function in the thickness direction, this yields a complete three-dimensional representation of the shell. The proposed shell element is implemented in a standard finite element code using Bézier extraction. The formulation is verified using different benchmark tests. Copyright © 2013 John Wiley & Sons, Ltd.

1 INTRODUCTION

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 SOLID-LIKE SHELL FORMULATION
  5. 3 ISOGEOMETRIC FINITE ELEMENT DISCRETIZATION
  6. 4 NUMERICAL SIMULATIONS
  7. 5 CONCLUDING REMARKS
  8. APPENDIX A: STRAIN COMPONENTS
  9. APPENDIX B: VIRTUAL STRAINS AND DERIVATIVES
  10. REFERENCES

Isogeometric analysis (IGA) has recently received much attention in the computational mechanics community. The basic idea is to use splines, which are the functions commonly used in computer-aided design (CAD) to describe the geometry, as the basis function for the analysis rather than the traditional Lagrange polynomial functions [1, 2]. Originally, non-uniform rational B-splines (NURBS) have been used in IGA, but their inability to achieve local refinement has driven their gradual replacement by T-splines [3].

A main advantage of IGA is that the functions used for the representation of the geometry are employed directly for the analysis, thereby bypassing the need for a sometimes elaborate meshing process. This important feature allows for a design-through-analysis procedure, which yields a significant reduction of the time needed for preparation of the analysis model [2]. Indeed, the exact parametrization of the geometry can have benefits for the numerical simulation of shell structures, which can be very sensitive to imperfections in the geometry. Moreover, the higher-order continuity of the shape functions used in IGA allows for a straightforward implementation of shell theories, which require inline image continuity such as Kirchhoff–Love models [4, 5]. A Reissner–Mindlin shell formulation has been developed by Benson et al. [6] using NURBS basis functions. Although inline image continuity is then no requisite, good results and a high degree of robustness were reported for large deformation problems. In addition, the exact geometry description allows for an exact computation of the shell director [7].

A further benefit of basis functions that possess a higher degree of continuity is that the computation of stresses is vastly improved. In shell analysis, this can be particularly important when materially nonlinear phenomena such as damage, or delamination, which can occur in laminated spatial structures, are included in the analysis. In the latter case, the computation of an accurate three-dimensional stress field becomes mandatory, and solid-like shell elements become an obvious choice [8-10]. The latter class of shell elements is characterized by the absence of rotational degrees of freedom, which is convenient when stacking them, yet possess shell kinematics, and are rather insensitive to shear locking and membrane locking.

Herein, we will develop a solid-like shell element that is based on the isogeometric concept. It therefore combines the advantage of an accurate geometric description of the shell mid-surface and the advantages in terms of meshing of IGA with the three-dimensional stress representation of conventional solid-like shell elements. The formulation adopts NURBS (or T-spline) basis functions for the discretization of the shell mid-surface, whereas in the thickness direction, conventional Lagrange polynomials are used. The basic formulation is outlined in Section 2, whereas algorithmic and implementation aspects are discussed in Section 3. Section 4 contains a number of benchmark problems to assess the performance of the isogeometric solid-like shell formulation. So as not to obscure the comparison by effects that are not purely caused by the shell formulation itself, no allowance has been made for plasticity or damage, although locally the strains can be such that these effects could occur. Furthermore, the comparison is restricted to monolayer shells, the extension to delamination being envisioned in a subsequent contribution. The paper concludes with some observations on the efficiency of the use of basis functions with a high degree of smoothness for problems, which exhibit highly localized deformation modes such as wrinkling.

2 SOLID-LIKE SHELL FORMULATION

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 SOLID-LIKE SHELL FORMULATION
  5. 3 ISOGEOMETRIC FINITE ELEMENT DISCRETIZATION
  6. 4 NUMERICAL SIMULATIONS
  7. 5 CONCLUDING REMARKS
  8. APPENDIX A: STRAIN COMPONENTS
  9. APPENDIX B: VIRTUAL STRAINS AND DERIVATIVES
  10. REFERENCES

As noted in the Introduction, shell elements that are based on the Reissner–Mindlin theory, and that are commonly obtained by degenerating a solid, are less convenient for use in laminated spatial structures, which is due to the presence of rotational degrees of freedom. With a view on extending the isogeometric shell element that we develop herein to laminated shell structures, we have therefore taken the solid-like shell element proposed by Parisch [8] as point of departure, because this element only possesses displacement degrees of freedom. Moreover, an internal stretch term is added in this element, so that a quadratic term is obtained in the displacement field in the thickness direction. As a consequence, the normal strain varies linearly, which significantly reduces membrane locking and results in a full three-dimensional stress state. The original element by Parisch [8] has been extended in [9, 10] for use in laminated composites, including interlaminar delamination.

2.1 Kinematics

Figure 1 shows the reference and the current configurations of the element, and the kinematics in curvilinear coordinates. We describe the element kinematics through a linear combination of a pair of material points at the top and at the bottom surfaces of the element. Each point at the top or at the bottom surface of an element in the original configuration is labeled by its position vectors: Xt and Xb, respectively. The variables ξ and η are the local curvilinear coordinates in the two independent in-plane directions, and ζ is the local curvilinear coordinate in the thickness direction. The position of a material point in the undeformed configuration is written as a function of the three curvilinear coordinates:

  • display math(1)

where X0(ξ,η) is the projection of the point on the mid-surface of shell and D is the thickness director at this point. In conventional solid-like shell elements, they are obtained as:

  • display math(2)

In an isogeometric formulation, these quantities are computed directly, as we will show in the remainder of this paper. The position of the material point in the deformed configuration X(ξ,η) is related to X(ξ,η) via the displacement field ϕ(ξ,η,ζ) as:

  • display math(3)

where

  • display math(4)

In this relation, u0 and u1 are the displacement of X0 on the shell mid-surface and the motion of the thickness director D, respectively. The projection of the material point onto the mid-surface leads to:

  • display math(5)

and:

  • display math(6)

Conventionally, the displacements u0 and u1 are calculated as:

  • display math(7)
  • display math(8)

which is convenient for applying the boundary conditions, although not strictly necessary in an isogeometric approach. In Equation (4), u2 is the internal stretching of the element, which is colinear with the shell thickness director D in the deformed configuration. This quantity is expressed in terms of stretch degree of freedom, w, through:

  • display math(9)

In any material point, a local reference triad can be established. The covariant base vectors are then obtained as the partial derivatives of the position vectors with respect to the curvilinear coordinates Θ = [ξ,η,ζ]. In the undeformed configuration, they are defined as:

  • display math(10)

where (.),α denotes the partial derivative with respect to Θα. Eα is the covariant base vector defined on the mid-surface:

  • display math(11)

Similarly, in the deformed configuration, we have:

  • display math(12)
image

Figure 1. Geometry and kinematics of the shell in the undeformed and in the deformed configurations.

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By using Equations (10) and (12), the components Gij and Gij of the metric tensors in the undeformed and the deformed configuration, respectively, can be determined as:

  • display math(13)

Elaboration of the expressions leads to the following components of the metric tensor in the undeformed configuration:

  • display math(14)

The elaboration of the components Gij includes higher-order terms that are up to the fourth order in the thickness coordinate ζ and derivatives of the stretch u2 with respect to the coordinates ξ and η. Neglecting these terms results in [8]:

  • display math(15)
  • display math

By using Equations (14) and (15), we can rewrite the components of the metric tensors as:

  • display math(16)

where inline image and inline image correspond to the constant terms in Equations (14) and (15), whereas inline image and inline image represent the linear terms. The contravariant base vectors needed for the calculation of the strains can be derived as:

  • display math(17)

with G symbolically denoting the metric tensor in the undeformed configuration. The contravariant base vectors denoted by the super script j are related to the corresponding base vectors on the mid-surface via the so-called shell tensor inline image as:

  • display math(18)

where inline image is the Kronecker delta and inline image denotes the mixed variant metric tensor, which is calculated with the covariant and contravariant tensors defined in Equation (16):

  • display math(19)

which is only nonzero when the undeformed shell is curved. The volume of the element in the undeformed configuration is evaluated using the metric tensor Gij in the following manner:

  • display math(20)

2.2 Strain measure

The Green–Lagrange strain tensor γ is defined conventionally in terms of deformation gradient F:

  • display math(21)

where I is the unit tensor. The deformation gradient can be written in terms of the base vectors as:

  • display math(22)

which leads to following representation of the Green–Lagrange strain tensor:

  • display math(23)

Substituting Equation (18) into this relation yields:

  • display math(24)

Next, the strain tensor can be expressed in the membrane mid-surface strain, εij, and the bending strain, ρij, as follows:

  • display math(25)

The strain components εij and ρij can be found in Appendix A.

2.3 Virtual work and linearization

In a Total Lagrangian formulation, the internal virtual work is expressed in the reference configuration Ω0:

  • display math(26)

with S the Second Piola–Kirchhoff stress tensor. The components of the virtual membrane strain, δεij, and those of the virtual bending strain, δρij, are given in Appendix B.

The resulting system of nonlinear equations is typically solved in an incremental-iterative manner, which requires computation of the tangential stiffness matrix. This quantity is obtained by linearizing the internal virtual work, Equation (26):

  • display math(27)

with δγ and inline image defined in Appendix B.

In solid-like shell elements, the stresses are computed using a three-dimensional constitutive relation. Assuming small strains, a linear relation between the rates of the Second Piola–Kirchhoff stress tensor and the Green–Lagrange strain tensor can be adopted:

  • display math(28)

where C is the material tangential stiffness matrix. Substitution of this identity into Equation (27) gives:

  • display math(29)

3 ISOGEOMETRIC FINITE ELEMENT DISCRETIZATION

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 SOLID-LIKE SHELL FORMULATION
  5. 3 ISOGEOMETRIC FINITE ELEMENT DISCRETIZATION
  6. 4 NUMERICAL SIMULATIONS
  7. 5 CONCLUDING REMARKS
  8. APPENDIX A: STRAIN COMPONENTS
  9. APPENDIX B: VIRTUAL STRAINS AND DERIVATIVES
  10. REFERENCES

In this section, we review some basic concepts of IGA. Next, the Bézier extraction technique will be outlined. This method is utilized to make a finite element data structure of the spline basis functions.

3.1 Fundamentals of B-splines and NURBS

A B-spline is a piecewise polynomial curve composed of linear combinations of B-spline basis functions:

  • display math(30)

where P is the order and N is the number of the basis functions. The Ni,p(ξ) represents a B-spline basis function and the coefficients Pi are points in space, referred to as control points. B-splines are defined over a knot vector, Ξ, which is a set of nondecreasing real numbers representing coordinates in the parameter domain:

  • display math(31)

Parametric coordinates, ξi, divide the B-spline into sections. In one dimension, the B-spline basis functions are defined recursively using the Cox-de Boor relation [11, 12]. By using tensor products, B-spline surfaces can be constructed using two knot vectors Ξ = {ξ1,ξ2, … ,ξn + p + 1}, H = {η1,η2, … ,ηm + q + 1}and an n × m set of control points Pi,j known as the control net. By defining one-dimensional basis function Ni,p and Mj,p over these two knot vectors, the B-spline surface is constructed as:

  • display math(32)

A drawback of B-splines is their inability to represent engineering objects such as conical sections exactly. For this reason, NURBS, which encapsulate B-splines and can represent such objects exactly, have become the standard in CAD. NURBS are defined by augmenting each control point with a weight Wi > 0 as Pi = (xi,yi,zi,Wi). Such a point can be represented with homogeneous coordinates inline image in a projective inline image space. Accordingly, NURBS basis functions are defined as:

  • display math(33)

where inline image is the weighting function. Note that there is no summation implied over the repeated index α, and that a B-spline is recovered when all the weights are equal. The NURBS surfaces are constructed by a tensor product of the one-dimensional functions as mentioned in Equation (32), similar to B-splines.

3.2 Bézier extraction

As noted in the previous section, the parametric coordinates in a knot vector ξi divide the parameter domain into elements. Similar to the FEM, these elements, which refer to the knot intervals {ξi,ξi + 1}with a positive length, allow for piecewise integration using quadrature rules. On the other hand, basis functions Ni,p, have a local support over a knot interval {ξi,ξi + p + 1}, which means that each element supports different basis functions, see Figure 2. This is at variance with the FEM where numerical integration is done on a single parent element. To blend IGA into existing finite element computer programs, we use Bézier elements and Bézier extraction operators [13] to provide a finite element structure for B-splines, NURBS, and T-splines representations [14].

image

Figure 2. B-spline basis function plotted over [ − 1,1]. The basis functions are different per element, which is in contrast with standard finite elements.

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In general, a degree P Bézier curve is defined by a linear combination of p + 1 Bernstein basis functions B(ξ) [15]. Similar to the B-splines, by having an appropriate set of control points, a Bézier curve is written as C(ξ) = PTB. The Bézier extraction operator maps a piecewise Bernstein polynomials basis onto a B-spline basis. This transformation makes it possible to use Bézier elements as the finite element representation of B-splines, NURBS, or T-splines.

The extraction operator can be obtained by means of knot insertion. Consider a knot vector Ξ and a set of control points inline image. By inserting a knot value inline image in the knot vector, a new set of control points needs to be calculated. This new set can be related to the initial set of control points via:

  • display math(34)

This relation ensures that the parametrization is not changed when an existing knot value is repeated, see [13, 15] for algorithms to determine the operator C1. The knot insertion process is repeated until all interior knots of the knot vector have a multiplicity equal to P, with P the order of the original spline defined over the knot vector Ξ. Next, the complete set of new control points inline image, with m = nep + 1, is obtained as:

  • display math(35)

Again, parametrization remains unchanged upon the insertion of the additional knots. Hence, according to Equation (30) and by using Equation (35), it is expressed as:

  • display math(36)

Because P is arbitrary, the refined basis functions B are related to the original basis functions N via:

  • display math(37)

Hence, every original basis function can be expressed as a linear combination of the Bernstein polynomials. By defining the operators Le and inline image to select the basis function Ne and Be, which are defined over the elements, we have:

  • display math(38)

In this way, the element extraction operator can be elaborated as:

  • display math(39)

As it can be observed from Figure 3, the Bézier extraction operator of an element Ce maps a piecewise Bernstein polynomial basis onto a B-spline basis.

image

Figure 3. Schematic representation of the Bézier extraction operator.

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The Bézier extraction operator for multivariate B-splines and NURBS can be computed by exploiting their tensor product structure, see [13] for details. For a detailed discussion of Bézier extraction for T-splines, for which a global tensor product structure is absent, see [14]. From the element extraction operator, Bézier elements and the global Bézier mesh can be constructed, see Figure 4.

image

Figure 4. Schematic representation of Bézier extraction: (a) a Bézier element corresponding to a third-order NURBS; (b) the Bézier element is mapped to the physical mesh using the Bézier extraction operator.

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3.3 Isogeometric finite element implementation

The solid-like shell element as developed in [8-10] can formulated as an 8-node or as a 16-node element. In both cases, a linear distribution of the internal stretch is assumed, so that only four internal degrees of freedom, located at the four corners of the mid-surface of the element are necessary. In this formulation, the projected displacements at the top and at the bottom surfaces, ut and ub, are constructed using the degrees of freedom of the top and the bottom nodes, respectively. For example, the displacement field for a quadrilateral 16-node element can be constructed using eight biquadratic shape function used for the both top and the bottom nodes together with four bilinear shape functions that are used for the discretization of the stretching.

We depart from the solid-like shell formulation outlined in Section 2, but we now only model the mid-surface of the shell. Accordingly, the three-dimensional representation of the shell reduces to a two-dimensional description using Bézier elements, where the geometric and the kinematic quantities are approximated by NURBS functions. In a Bézier mesh, each control point Pi contains a vector of degrees of freedom Φi, as follows:

  • display math(40)

where ax, ay, az denote the displacement components and aw is the stretch degree of freedom. The superscripts b and t correspond to the top and bottom surfaces of the shell, respectively, and np is the number of control points in an element. For simplicity, we rewrite the vector Φi as:

  • display math(41)

where inline image corresponds to the bottom and the top degrees of freedom, defined on the mid-surface of the shell. In a Bézier element, the stretching is interpolated at each control point instead of varying linearly. To calculate the displacement terms in Equations (7) and (8), we construct the matrix inline image for a Bézier element E as:

  • display math(42)

with Ni,p(ξ,η) the NURBS basis functions of order P. The displacements of the shell mid-surface and the displacement of the thickness director, u0 and u1, respectively, are obtained from Equations (7) and (8), and can be discretized as:

  • display math(43)

where N0 and N1 are defined as:

  • display math(44)
  • display math(45)

and the vector a contains the translational degrees of freedom on the mid-surface:

  • display math(46)

The stretch is interpolated with the same spline basis formulations as those for the in-plane displacement field:

  • display math(47)

so that the interpolation matrix for the stretching can be written as:

  • display math(48)

The derivatives of the displacement vectors u0 and u1 with respect to the parametric coordinates ξ and η follow in a standard manner as:

  • display math(49)

with the matrices inline image and inline image containing the derivatives of the basis functions:

  • display math(50)

and

  • display math(51)

where:

  • display math(52)

3.4 Evaluation of internal force vectors and stiffness matrices

For the evaluation of the tangential stiffness matrices, we first define the virtual strain vector:

  • display math(53)

The virtual strain vector can be decomposed into components, as follow:

  • display math(54)

where the H matrices can be obtained from the strain variation, Appendix B. Next, the variations of the displacement vectors are computed from the degrees of freedom associated to the control points via the B-spline basis functions:

  • display math(55)
  • display math(56)
  • display math(57)
  • display math(58)

Substituting these relations into Equation (54) relates the vector of virtual strains to the control points degrees of freedom:

  • display math(59)

with the matrices Bu and Bw defined as:

  • display math(60)
  • display math(61)

Now, from the internal virtual work, Equation (26), the internal force vector is directly obtained as:

  • display math(62)

Next, we rewrite the linearized internal virtual work, Equation (29), in matrix form:

  • display math(63)

where K represents the stiffness matrix decomposed in a material part Kmat and a geometric part, Kgeom, as usual. From Equation (29), these matrices can be obtained as:

  • display math(64)

The geometric part is the stress-dependent part of the stiffness matrix and is obtained through the derivatives of the virtual strains, Appendix B.

3.5 Evaluation of the shell director

In the original solid-like shell formulation [8], the shell director D at the corners of an element were derived from the positions of the corresponding nodes on the top and the bottom surface of this element. The director in any arbitrary point on the mid-plane surface in the element followed from the interpolation scheme. As a result, the shell director was at best a linear or quadratic approximation. In the case of strongly curved shells, this may lead to significant errors.

In this paper, we calculate the shell director D directly by using the mid-surface base vectors Eα,α = 1,2, similar to [7] and identical to [5] (Chapter 15). The mid-surface vectors, which are formulated in Equation (11), are discretized with a B-spline interpolation scheme. As a result, these vectors represent exactly the in-plane directions of the shell and can be used to construct the out-of-plane director D, according to:

  • display math(65)

In this relation, t is the thickness of the shell. Note that in the current implementation, it is assumed that the thickness t is constant.

4 NUMERICAL SIMULATIONS

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 SOLID-LIKE SHELL FORMULATION
  5. 3 ISOGEOMETRIC FINITE ELEMENT DISCRETIZATION
  6. 4 NUMERICAL SIMULATIONS
  7. 5 CONCLUDING REMARKS
  8. APPENDIX A: STRAIN COMPONENTS
  9. APPENDIX B: VIRTUAL STRAINS AND DERIVATIVES
  10. REFERENCES

The isogeometric solid-like shell formulation is now verified and assessed through different benchmark tests. The NURBS-based CAD program Rhino has been used for modeling the geometry. Because the present formulation is based on the discretization of the mid-surface of the shell, only this surface has been modeled in Rhino. Subsequently, the Rhino model is converted to a data file that contains the information on the control points, the elements, and the Bézier extraction operator. This data file is exported to a code that works as the preprocessor, so we are able to select the control points, elements, and edges for the application of the loads and other boundary conditions. A finite element compatible input file is then generated, which is directly exported to the finite element code. A 4 × 4 integration scheme has been used in all benchmark tests, see [16] for a detailed discussion on integration schemes.

4.1 Clamped cylindrical shell

The first example concerns a clamped cylindrical shell subject to bending, Figure 5. The cylinder has a radius R = 10mm, a length L = 100mm, a thickness t = 0.5mm, a Young's modulus E = 2 × 107N ∕ mm2, and a Poisson's ratio ν = 0.3. It is clamped at one end and subjected to a vertical load P at the other end. The exact geometry of the cylinder mid-surface has been constructed using second-order or third-order NURBS basis functions.

image

Figure 5. Cylindrical shell modeled with four solid-like shell Bézier (SLSBEZ) elements.

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The simulation has been carried out for various Bézier meshes, each consisting of M × N elements, where M and N are the number of elements in the longitudinal and the circumferential directions, respectively. For the validation of this test, a linear analytical solution obtained from beam theory is used, where the deflection of a beam δ classically reads PL3 ∕ 3EI.

The results obtained for both orders of interpolation are shown in Tables 1 and 2. In these tables, the deflection is normalized with respect to the analytical solution obtained for a load P = 800 N.

Table 1. Normalized deflection, p = 2.
Mesh1 × 41 × 81 × 162 × 44 × 4
Deflection0.86820.87110.87190.99011.032
Table 2. Normalized deflection, p = 3.
Mesh1 × 41 × 81 × 162 × 44 × 4
Deflection1.0121.0171.0361.0281.034

Table 1 shows that, when using second-order basis functions, a response is obtained for the three coarser meshes that is slightly stiffer than the analytical beam solution. Upon mesh refinement, an increasingly softer response is computed, and for the finest mesh, the response is even softer than the beam solution. Indeed, the kinematic assumptions that underlie the beam solution are slightly different from the kinematics on which the present shell model is based, which explains that the analytical solution is not retrieved exactly. This is even more pronounced when using third-order basis functions. Then, all results are a bit softer than the beam solution, Table 2.

4.2 Pinched hemispherical shell with hole

A pinched hemisphere with a hole at the top has been used extensively as a benchmark problem for shell analysis to test the ability to describe nearly inextensional bending modes [17-19]. The geometric parameters and the material properties employed in this test are summarized in Table 3. The shell is subjected to two opposite point loads. The bottom circumferential edge of the hemisphere is free. Because of the symmetry, only a quarter of the shell needs to be modeled. The symmetric boundary conditions are applied by constraining the displacement degrees of freedom in the normal direction of the symmetry plane. The mesh and the applied boundary conditions are shown in Figure 6. ABAQUS has been used to generate an alternative, traditional finite element solution, using a 16 × 16 mesh consisting of so-called S4R shell elements.

Table 3. Geometric parameters and material properties for the pinched hemisphere.
Radius RThickness tYoung's modulus EPoisson's ratio ν
10.00.046.825 × 1070.3
image

Figure 6. The mesh for a quarter model and the boundary conditions.

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Figure 7 shows the load-displacement curves of the pinched hemisphere that have been obtained for different meshes. A plot of the deformed configuration and of the Von Mises stresses is given in Figure 8. The results from a coarse mesh with 4 × 4 solid-like shell Bézier (SLSBEZ) elements exhibit an overly stiff behavior. Increasing the number of elements to 8 × 8 results in a significant softer response, whereas refining the mesh once more – to 16 × 16 elements – leads to results that are close to the traditional finite element solution (using S4R elements). The graph also shows the results from a 8 × 8 mesh of standard solid-like shell elements, which is between those obtained with the medium mesh and with the fine mesh of SLSBEZ elements.

image

Figure 7. Diagram of the load P versus the displacement at point A for the pinched hemisphere.

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image

Figure 8. Deformed configuration of the pinched hemisphere. Results for 16 × 16 SLSBEZ elements.

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To make a first comparison between the results, we list the number of degrees of freedom for the used meshes in Table 4. The number of degrees of freedom is not necessarily proportional to the CPU time that is required to carry out a computation. For instance, the process of building the stiffness matrix using Bézier extraction can be more time-consuming than for traditional finite elements, but the smoother stress fields obtained using the SLSBEZ elements may result in a faster convergence of the iterative process that is utilized to solve the nonlinear set of equations. Moreover, the use of different finite element packages and different (linear) solvers to generate the results of Figure 7 make a direct comparison in terms of CPU time impossible. Nevertheless, it is clear that in this example, elements that use splines as basis functions seem to be about as efficient as traditional finite elements that employ Lagrangian polynomials. This is at variance with results reported for other applications, and may be due to the fact that the inline image continuity at the element boundaries of traditional finite elements may actually facilitate the capturing of deformation patterns that exhibit a locally strong curvature.

Table 4. Number of DOFs for different meshes employed in pinched hemispherical shell test.
Mesh4 × 4 SLSBEZ8 × 8 SLSBEZ8 × 8 SLS16 × 16 SLSBEZ16 × 16 S4R
Number of DOFs343847160625271445

4.3 Pinched cylinder with free ends

The pinched cylinder with free ends shown in Figure 9 is used next to assess the element performance. The parameters that define the geometry and the material properties are summarized in Table 5. The cylinder has free edges at the ends, and it is loaded by two centrally located diametrically opposed point forces, which pull in the outward direction. Because of symmetry considerations, only one-eight of the cylinder needs to be modeled.

image

Figure 9. Pinched cylinder with free ends.

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The initial response is dominated by the bending stiffness, which induces large displacements at relatively low load levels. This changes into a very stiff response when the displacement becomes larger. Finite rotations occur afterwards, thus making the pinched cylinder with free ends a challenging test for element performance [20-22].

Table 5. Geometry parameters and material properties for the pinched cylinder with free ends.
Radius RLength LThickness tYoung's modulus EPoisson's ratio ν
4.95310.350.09410,5000.3125

Figure 10 shows the load-displacement curve obtained for different meshes of SLSBEZ elements, for standard solid-like shell elements, and for the S4R shell elements implemented in ABAQUS. The magnitude of the load is that for the complete cylinder and the displacement is measured at point A. Figure 11 shows the deformed configuration and Von Mises stresses for a half model.

image

Figure 10. Load-displacement diagram of pinched cylinder with free ends.

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image

Figure 11. Deformed configuration of pinched cylinder meshed with 8 × 8 SLSBEZ elements.

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From Figure 10, it is inferred that all results are very close, with exception of the solution for the 8 × 8 mesh composed of SLSBEZ elements, which is slightly stiffer. Table 6 gives the number of degrees of freedom for each calculation and seems to indicate a somewhat better efficiency for finite elements equipped with standard polynomials. This confirms the results obtained for the example of the pinched hemisphere, obviously subject to the same reservations. It must be noted, however, that efficiency is not the only criterion. For instance, the availability of a fully three-dimensional stress field in solid-like shell elements makes this formulation richer and superior to standard shell elements when material nonlinearities such as plasticity, damage, and delamination are included in the analysis. Furthermore, a comparison that singles out a single displacement degree of freedom may not be representative and obscures the full picture. For instance, the stress prediction using IGA is vastly improved compared with standard finite elements, which has advantages, again particularly for materially nonlinear analysis.

Table 6. Number of DOFs for different meshes employed for the pinched cylinder with free ends.
Mesh8 × 8 SLSBEZ16 × 16 SLSBEZ16 × 8 SLS16 × 8 S4R
Number of DOFs84725272598765

4.4 Pinched cylinder with rigid diaphragm

The problem of a pinched cylinder with a rigid diaphragm (see Figure 12) at the ends has been studied by several authors [23-25] to test the convergence behavior and nonlinear performance of shell elements. Because large rotations occur, the problem provides a test for the finite rotation capability of the shell formulation. The dimensions and the material properties are shown in Table 7. The cylinder is loaded by two centrally located, diametrically opposed point forces P, which push inwards. By using symmetry, only one-eighth of the structure needs to be modeled.

image

Figure 12. Pinched cylinder with rigid diaphragm.

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Table 7. Geometry parameters and material properties for the pinched cylinder with a rigid diaphragm at the ends.
Radius RLength LThickness tYoung's modulus EPoisson's ratio ν
100200130,0000.3

Figure 13 shows results for a coarse, uniform mesh of 16 × 16 SLSBEZ elements and for a reference mesh of 40 × 40 S4R shell elements. The coarse mesh with SLSBEZ elements shows a far too stiff behavior, which can be explained by the fact that it cannot well capture the effect of local wrinkles. As a next step, results obtained through a local mesh refinement are presented in Figure 14. As the wrinkles emerge close to the left edge of the structure, this part of the model is locally refined using T-splines. The load-displacement curve shows an improvement, but it is not smooth and is still stiffer than the reference shell solution. A second mesh refinement is carried out, which results in the load-displacement curve shown in Figure 15, which is in good agreement with the reference solution using S4R shell elements. To compare the performance of the SLSBEZ element with that of the underlying solid-like shell element, we repeat the simulation by using a fine mesh of 80 × 80 standard solid-like shell elements, see also Figure 15. The result clearly shows a stiffer response than that of the finest mesh of SLSBEZ elements (see Table 8).

image

Figure 13. Load-displacement diagram of pinched cylinder with rigid diaphragm. First mesh: 256 SLSBEZ elements.

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image

Figure 14. Load-displacement diagram of pinched cylinder with rigid diaphragm. The right part of the figure shows the first local mesh refinement using T-splines.

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image

Figure 15. Load-displacement diagram of pinched cylinder with rigid diaphragm. The right part of the figure shows the second local mesh refinement using T-splines.

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Table 8. Number of DOFs for different meshes employed for the pinched cylinder with a rigid diaphragm at the ends.
Mesh847 SLSBEZ80 × 80 SLS40 × 40 S4R
Number of DOFs658739,3668405

Figure 15, finally, presents the deformed configuration. The plot of the Von Mises stresses shows that locally very high stresses are computed, which normally will lead to plasticity or damage. Material nonlinearities have not been included in the present analysis, however, so as not to mix the results regarding the element performance with effects that can emanate from the integration of the constitutive relation (also see Figure 16 for more details).

image

Figure 16. Deformed configuration of pinched cylinder with rigid diaphragm.

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5 CONCLUDING REMARKS

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 SOLID-LIKE SHELL FORMULATION
  5. 3 ISOGEOMETRIC FINITE ELEMENT DISCRETIZATION
  6. 4 NUMERICAL SIMULATIONS
  7. 5 CONCLUDING REMARKS
  8. APPENDIX A: STRAIN COMPONENTS
  9. APPENDIX B: VIRTUAL STRAINS AND DERIVATIVES
  10. REFERENCES

A solid-like shell element has been formulated that is based on the isogeometric concept. Spline basis functions (NURBS or T-splines) have been used to parametrize the mid-surface, whereas a linear Lagrange shape function has been employed in the thickness direction. In this manner, a complete three-dimensional representation of the shell is obtained. The shell formulation combines the advantages of the solid-like shell formulation, such as the full, three-dimensional stress and strain representation that allows for the straightforward implementation of constitutive relations such as plasticity or damage, with the advantages of IGA, including the exact description of the geometry, the use of the design-through-analysis concept, and the accurate prediction of stress fields. The latter property is also beneficial for the prediction of the onset of plasticity, damage, or interlaminar delaminations due to high transverse stresses.

In this study, the performance of the isogeometric solid-like shell element has been assessed by means of a number of benchmark tests that involve geometric nonlinearity, comparing the new element with conventional solid-like shell elements and with a standard shell element that is available in a commercially available code. Although the spline-based solid-like shell element tends to perform slightly better than the conventional solid-like shell element, the observations regarding its performance vis-à-vis the standard shell element are inconclusive, with the standard shell element tending to perform better. It should be emphasized, however, that the comparisons are done, as usual, on a global basis, that is, by comparing load-displacement curves. Different conclusions may be reached if the predictions of the local stress fields are compared, or more generally, if a local property is taken as benchmark property, which benefits from the higher smoothness of the spline basis functions. Indeed, the analyzed examples suggest that the smoothness of spline functions can be counterproductive in obtaining a fast convergence upon mesh refinement for problems that involve highly localized deformations like local buckling, or wrinkling. This holds a fortiori when NURBS are used, but is ameliorated through the use of T-splines, since they allow for an effective local mesh refinement.

To keep the comparison as objective as possible, we have taken into account no materially nonlinear effects, although in certain cases, the stresses became so high that plasticity or damage would have occurred. By ignoring these effects, however, the element performance was not diluted by errors in the integration of the constitutive relation, nor by possible effects of a discontinuity, which for instance enters at an elasto-plastic boundary.

APPENDIX B: VIRTUAL STRAINS AND DERIVATIVES

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 SOLID-LIKE SHELL FORMULATION
  5. 3 ISOGEOMETRIC FINITE ELEMENT DISCRETIZATION
  6. 4 NUMERICAL SIMULATIONS
  7. 5 CONCLUDING REMARKS
  8. APPENDIX A: STRAIN COMPONENTS
  9. APPENDIX B: VIRTUAL STRAINS AND DERIVATIVES
  10. REFERENCES

The virtual strain components are elaborated as:

  • display math(B.1)

with inline image.

The derivatives of the virtual strains, are given by:

  • display math(B.2)

REFERENCES

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 SOLID-LIKE SHELL FORMULATION
  5. 3 ISOGEOMETRIC FINITE ELEMENT DISCRETIZATION
  6. 4 NUMERICAL SIMULATIONS
  7. 5 CONCLUDING REMARKS
  8. APPENDIX A: STRAIN COMPONENTS
  9. APPENDIX B: VIRTUAL STRAINS AND DERIVATIVES
  10. REFERENCES
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