# High order exact geometry finite elements for seven-parameter shells with parametric and implicit reference surfaces

- 379 Downloads
- 1 Citations

## Abstract

We present high order surface finite element methods for the linear analysis of seven-parameter shells. The special feature of these methods is that they work with the exact geometry of the shell reference surface which can be given parametrically by a global map or implicitly as the zero level-set of a level set function. Furthermore, a special treatment of singular parametrizations is proposed. For the approximation of the shell displacement parameters we have implemented arbitrary order hierarchical shape functions on quadrilateral and triangular meshes. The methods are verified by a convergence analysis in numerical experiments.

## Keywords

Shells Surface finite elements Exact geometry Implicit geometry Higher order## 1 Introduction

Due to their efficient load-carrying capabilities, shell structures enjoy widespread use in a variety of engineering applications. Therefore, a huge amount of research work has been devoted to the development of shell models (e.g. [11, 15]), their formal justification (e.g. [12, 13, 14]), as well as to the realization of numerical methods (e.g. [8, 9]). It is well known that shell structures are sensitive to geometric imperfections [30]. Even worse, the approximation of the geometry leads to wrong solutions in some situations, see the plate paradox investigated e.g. in [2]. Furthermore, e.g., in contact problems an exact description of smooth geometries is of interest. Among others, these examples motivated us to develop shell finite element methods based on the exact geometry. Therefore, the literature review concentrates on geometry representation.

Usually, the exact geometry is approximated. In the simplest case, planar facet elements are deployed. However, those elements can only poorly represent curved structures and are not reliable due to a missing bending–stretching coupling on the element level. In order to represent the geometry more accurately high order (often quadratic) shape functions are used. To our best knowledge, exact geometry methods for shell analysis are restricted to the case where the exact geometry is parametrically defined. A finite element method which allows for arbitrary parametrizations is described in [1]. Therein, the field approximation based on Lagrange elements is applied to a seven-parameter shell theory considering geometrical nonlinearities and functionally graded shells. However, the actual computation of the arising integrals is carried out by symbolic algebra subroutines written in MAPLE. In [31], an exact geometry method based on the blending function method is presented. Furthermore, in the work [29] a finite element method for geometrically nonlinear problems is developed. Therein, the exact geometry of the shell is captured by an initial deformation of a flat reference configuration. However, since most CAD systems use NURBS functions, it is reasonable to restrict the input parametrizations for the shell analysis to NURBS [10]. This has the advantage that the parametrization is given as a product of basis functions and coefficients. Thus, derivatives are easily computed if the derivatives of the basis functions are known. Following the concept of Isogemetric Analysis, shell finite elements based on different shell models were proposed in [5, 23, 24, 28] among others. Therein, the reference surface is described by NURBS, just as the field approximation.

Shell problems can also be seen as one application of the more general concept of partial differential equations defined on surfaces. We mention [17] for an overview of related finite element methods. To our best knowledge the first exact geometry method for the Laplace–Beltrami operator on implicitly defined surfaces was presented in [16]. Therein, the exact geometry of closed smooth surfaces is parametrized over a space triangulation by means of the closest point projection. Recently, a directional mapping based on predefined search directions was considered to construct high order geometry approximations [20, 25]. In [22], the search direction has been tailored such that the exact geometry of smooth surfaces with boundaries is available in the finite element analysis.

In the present paper we consider shells with parametrically and implicitly defined reference surfaces. For both cases exact geometry finite element methods are presented. For implicitly defined shells the reference surface is parametrized over a space triangulation following the developments presented in [22]. Thus, the implicit setting is reformulated to the parametrized setting. The curvature of the reference surface is necessary within the considered variational formulation, and so are the second order partial derivatives of the parametrization. In the case of a parametrically defined surface they are obtained by automatic differentiation based on hyper-dual numbers [19]. For the case of an implicitly defined surface, we derive a formula such that only the second order derivatives of the level-set function \(\phi \) are needed. As a shell model we use a displacement based seven-parameter model accounting for stretching, bending, shear deformations, and through-the-thickness stretching (see [1, 7, 15]). We restrict ourselves to a linear elastic analysis, i.e. small displacements, small rotations, small strains, and Hooke’s law. The discretization of the shell deformation is done by means of high order hierarchical \(H^1\)-conforming shape functions. Our implementation allows for arbitrary polynomial orders. Since our formulation is displacement based, various locking phenomena (membrane, shear and thickness locking) occur, which can be seen in the examples in Sect. 4. Therefore, the low order methods are very inefficient. Albeit the presence of locking effects, the proposed methods converge to correct solutions. Using high order shape functions reduces the locking effects and offers high convergence rates. Nevertheless, in many practical examples this approach might be not very efficient, see e.g. [29] for a quadratic element with a efficient displacement based formulation. As a further novelty, we propose a strategy for the modification of the shape functions to tackle singular parametrizations where the determinant of the metric vanishes on some part of the geometry. Following a similar strategy developed in the context of Isogeometric Analysis [33], we modify the ansatz space by combining and skipping basis functions.

## 2 Differential geometry and shell model

In this section, we recall the differential geometry of thin-walled structures and introduce the displacement based seven-parameter shell model. First, the geometry of the reference surface is presented. Then, the reference surface is extended to the three-dimensional shell volume, for which we present the geometric relations. In a next step, the three-dimensional shell problem is introduced. Finally, the kinematics are restricted to a seven-parameter shell model. We remark that the differential geometry in the context of thin-walled structures is exhaustively discussed in [3] and [11], among others.

### 2.1 Reference surface

### 2.2 Geometry of the shell volume

*t*is the thickness of the shell. The geometric setting is illustrated in Fig. 1. The first two base vectors in the shell volume are related to the base vectors at the reference surface by

### 2.3 The 3D shell problem

### 2.4 Seven-parameter shell model

## 3 Finite element method

*p*, \((p+1)^2\) in-plane quadrature points are used on each element, whereas three quadrature points for the integration across the thickness are employed. In the following, \(\varPhi ^e\) denotes the local linear element mapping. The discretization of the seven parameters \(\left\{ \overset{(1)}{u_i}, \overset{(2)}{u_i}, \overset{(n)}{u} \right\} \in [H^1({\bar{U}})]^3\times [H^1({\bar{U}})]^3\times H^1({\bar{U}})\) introduced in (15) is done by means of

### 3.1 FEM for shells defined by singular parametrizations

- 1.
All vertex-based shape functions related to the vertices on \(\theta ^1 = 0\) are added up to one single shape function.

- 2.
All edge-based shape functions related to the edges on \(\theta ^1 = 0\) are removed, i.e. the respective degrees of freedom are constrained to zero in the implementation.

- 3.
No modification of the cell-based shape functions is made.

### 3.2 FEM for implicitly defined shells

*a*in the following implicit way

*a*is implicitly defined because \(r(\mathbf{x})\) is not explicitly known and has to be computed in each evaluation of

*a*such that \(\phi (a(\mathbf{x})) = 0\). We specify the search directions in (25) as follows. Let \(\mathcal {V}\) denote the set of all vertices of \(\mathcal {T}_h\). We set

In order to clarify the explanations on the implicit method, we remark that the space triangulation represents an intermediate step in the method and does not define the geometry. The key ingredient of the implicit method is the mapping *a* defined in (25), which maps the space triangulation to the exact geometry. However, this mapping is only implicitly defined and a one dimensional root finding problem has to be solved for the evaluation. This is no problem because the integrals are approximated by quadrature. Thus, only a point-wise evaluation is necessary.

## 4 Numerical results

### 4.1 Verification example for a parametrically defined shell

*h*is a characteristic element length.

### 4.2 Verification example for an implicitly defined shell

*d*is the signed distance function of the torus. In order to come up with a manufactured solution, we construct a displacement field with respect to Cartesian coordinates. This displacement field should fulfill the kinematic assumptions introduced in (15). We note that the closest point projection

Again, we study the convergence rates under uniform mesh refinement. We remark that, because of the mapping of \({\bar{\varOmega }}_h\) to \({\bar{\varOmega }}\) used in the method, it is sufficient to subdivide the triangles of \(\mathcal {T}_h\) in a refinement step. Thus, for all refinement levels \({\bar{\varOmega }}_h\) is the one and the same surface. The results of the convergence study are given in Fig. 7. We observe optimal convergence rates.

### 4.3 Pinched hemisphere

Furthermore, we have solved the pinched hemisphere with the implicit formulation. The triangulations shown in Fig. 9 have been used. As an input we have given the four element triangulation where all vertices are on the reference surface. All other meshes have been obtained by uniform refinement. In contrast to a classical method, there is no need to better approximate the geometry by these triangulations, since they are mapped to the exact geometry within the method. In Table 3 the computed radial displacements at the loaded points are given. These results are similar to the results obtained by the parametric method given in Table 1.

### 4.4 Scordelis-Lo roof

Radial displacement at the loaded points of the pinched hemisphere

Number of elements | 16 | 64 | 256 |
---|---|---|---|

Linear | 0.0000039 | 0.0000112 | 0.0000373 |

Quadratic | 0.0000215 | 0.0001765 | 0.0026473 |

Cubic | 0.0001254 | 0.0203645 | 0.0823567 |

Quartic | 0.0344561 | 0.0868342 | 0.0921721 |

Quintic | 0.0591080 | 0.0919079 | 0.0924264 |

Sextic | 0.0915553 | 0.0923272 | 0.0924901 |

Septic | 0.0917929 | 0.0924269 | 0.0925234 |

Octic | 0.0923605 | 0.0924707 | 0.0925471 |

Sixth smallest eigenvalue of the stiffness matrix of the pinched hemisphere

Number of elements | 16 | 64 | 256 |
---|---|---|---|

Linear | 1.4e\(+\)05 | 1.5e\(+\)04 | 1.1e\(+\)03 |

Quadratic | 1.1e\(+\)03 | 2.2e\(+\)01 | 3.8e−01 |

Cubic | 1.8e\(+\)01 | 1.5e−01 | 1.0e−03 |

Quartic | 1.7e−01 | 2.2e−04 | 2.4e−07 |

Quintic | 1.0e−03 | 2.7e−07 | 3.9e−08 |

Sextic | 1.1e−05 | 4.7e−07 | 1.8e−07 |

Septic | 8.8e−07 | 6.5e−07 | 8.8e−08 |

Octic | 7.6e−06 | 8.9e−08 | 4.4e−08 |

Radial displacement at the loaded points of the pinched hemisphere

Number of elements | 4 | 16 | 64 | 256 |
---|---|---|---|---|

Linear | 0.0000011 | 0.0000035 | 0.0000134 | 0.0000526 |

Quadratic | 0.0000062 | 0.0000398 | 0.0005833 | 0.0075576 |

Cubic | 0.0000322 | 0.0011217 | 0.0262481 | 0.0877841 |

Quartic | 0.0002854 | 0.0125512 | 0.0878863 | 0.0923755 |

Quintic | 0.0016276 | 0.0607160 | 0.0922467 | 0.0924516 |

Sextic | 0.0058856 | 0.0894647 | 0.0924056 | 0.0924764 |

Septic | 0.0224888 | 0.0920513 | 0.0924452 | 0.0924957 |

Octic | 0.0572128 | 0.0923485 | 0.0924647 | 0.0925141 |

We use the function \(\kappa \) in order to capture the boundary layer. It has the properties \(\kappa (0)=0\), \(\kappa (1)=1\), \(\kappa '(0) = \kappa '(1) = b\) and is illustrated in Fig. 11. The parameter space is given by \(\theta ^1\in [0,1]\) and \(\theta ^2\in [0,1]\). In Fig. 10, a \(16\times 16 = 256\) element mesh mapped to the real space is illustrated.

*A*, which is located in the middle of one free edge and on the mid-surface. We remark that the vertical displacement varies considerably trough-the-thickness. The results for different ansatz orders and meshes are given in Table 4. It is evident that the low order methods are affected by locking. The results obtained with linear ansatz functions are far from the converged solution \(u_z={-0.3014}\). Raising the ansatz order reduces the locking phenomena. Without resorting to other techniques to reduce the locking, we advise to use at least quartic ansatz functions. In [27], a reference value \(u_z={-0.3024}\) for the vertical displacement at point

*A*is reported. For a shell model based on equivalent seven-parameter kinematics, \(u_z={-0.3008}\) is computed in [18]. Therefore, our results are in accordance with the values found in literature.

Vertical displacement at point A of the Scordelis-Lo roof

Number of elements | 4 | 16 | 64 | 256 |
---|---|---|---|---|

Linear | \(-\) 0.0026073 | \(-\) 0.0016144 | \(-\) 0.0044508 | \(-\) 0.0126987 |

Quadratic | \(-\) 0.0019732 | \(-\) 0.0305159 | \(-\) 0.1354229 | \(-\) 0.2741197 |

Cubic | \(-\) 0.0301026 | \(-\) 0.2470338 | \(-\) 0.2968267 | \(-\) 0.3012622 |

Quartic | \(-\) 0.1675085 | \(-\) 0.2967069 | \(-\) 0.3012862 | \(-\) 0.3014015 |

Quintic | \(-\) 0.2888778 | \(-\) 0.3012049 | \(-\) 0.3013835 | \(-\) 0.3014021 |

Sextic | \(-\) 0.2979929 | \(-\) 0.3013161 | \(-\) 0.3014014 | \(-\) 0.3014026 |

Septic | \(-\) 0.3014056 | \(-\) 0.3013603 | \(-\) 0.3014014 | \(-\) 0.3014026 |

Octic | \(-\) 0.3012498 | \(-\) 0.3013926 | \(-\) 0.3014021 | \(-\) 0.3014026 |

### 4.5 Gyroid

Vertical displacement \(u_z\) at the point \([ 2, {0.5}, {-0.25}]\)

Number of elements | 136 | 544 | 2176 |
---|---|---|---|

Linear | 0.0118106 | 0.0332828 | 0.1055900 |

Quadratic | 0.1694172 | 0.9005433 | 1.7152640 |

Cubic | 1.2274694 | 1.8430397 | 1.8793713 |

Quartic | 1.7745119 | 1.8788403 | 1.8808718 |

Quintic | 1.8726206 | 1.8804255 | 1.8811317 |

Sextic | 1.8784737 | 1.8809134 | 1.8812070 |

Septic | 1.8802114 | 1.8811026 | 1.8812299 |

Octic | 1.8807253 | 1.8811810 | 1.8812370 |

## 5 Conclusion

In this paper, high order finite element methods for shell analysis have been presented. The underlying shell model is a displacement based seven-parameter model. As a special feature, the methods incorporate the exact geometry of parametrically and implicitly defined reference surfaces. We have shown the capabilities of the methods in five examples. In order to assess the convergence behavior, the method of manufactured solutions has been utilized. In all numerical experiments, we observe optimal convergence rates in the asymptotic range.

In the present work we used a purely displacement based formulation. Thus, various locking phenomena reduce the efficiency of the method when using low order approximations. In order to reduce locking phenomena, we have resorted to high order shape functions (our implementation allows for arbitrary high order). As this might be not very efficient in many examples, it would be interesting to develop low order locking-free elements based on the exact geometry in future work.

Moreover, the use of a seven-parameter shell model allowed us to use \(H^1\)-conforming ansatz and test spaces. This might not be appropriate for other shell models like Kirchhoff–Love type models or Reissner–Mindlin type models. In the former, typically, \(H^2\)-conforming elements are needed, whereas in the latter elements are necessary providing a vector field tangential to the surface. The development of such elements for implicitly defined shells should receive further attention.

## Notes

### Acknowledgements

Open access funding provided by Graz University of Technology.

## References

- 1.Arciniega R, Reddy J (2007) Tensor-based finite element formulation for geometrically nonlinear analysis of shell structures. Comput Methods Appl Mech Eng 196(4):1048–1073MathSciNetCrossRefGoogle Scholar
- 2.Babuška I, Pitkäranta J (1990) The plate paradox for hard and soft simple support. SIAM J Math Anal 21(3):551–576MathSciNetCrossRefGoogle Scholar
- 3.Basar Y, Krätzig WB (1985) Mechanik der Flächentragwerke: Theorie, Berechnungsmethoden, Anwendungsbeispiele. ViewegGoogle Scholar
- 4.Belytschko T, Stolarski H, Liu WK, Carpenter N, Ong JS (1985) Stress projection for membrane and shear locking in shell finite elements. Comput Methods Appl Mech Eng 51(1):221–258MathSciNetCrossRefGoogle Scholar
- 5.Benson D, Bazilevs Y, Hsu MC, Hughes T (2010) Isogeometric shell analysis: the Reissner–Mindlin shell. Comput Methods Appl Mech Eng 199(5):276–289MathSciNetCrossRefGoogle Scholar
- 6.Bischoff M, Ramm E (1997) Shear deformable shell elements for large strains and rotations. Int J Numer Methods Eng 40(23):4427–4449CrossRefGoogle Scholar
- 7.Bischoff M, Ramm E (2000) On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation. Int J Solids Struct 37(46–47):6933–6960CrossRefGoogle Scholar
- 8.Bischoff M, Bletzinger KU, Wall W, Ramm E (2004) Models and finite elements for thin-walled structures. In: Stein E, de Borst R, Hughes T (eds) Encyclopedia of computational mechanics, chap 3, vol 2. Wiley Online Library, New York, pp 59–137Google Scholar
- 9.Chapelle D, Bathe KJ (2010) The finite element analysis of shells. Springer, BerlinzbMATHGoogle Scholar
- 10.Cho M, Roh HY (2003) Development of geometrically exact new shell elements based on general curvilinear co-ordinates. Int J Numer Methods Eng 56(1):81–115CrossRefGoogle Scholar
- 11.Ciarlet PG (2006) An introduction to differential geometry with applications to elasticity, vol 78. Springer, BerlinGoogle Scholar
- 12.Ciarlet PG, Lods V (1996) Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations. Arch Ration Mech Anal 136(2):119–161MathSciNetCrossRefGoogle Scholar
- 13.Ciarlet PG, Lods V (1996) Asymptotic analysis of linearly elastic shells. III. Justification of Koiter’s shell equations. Arch Ration Mech Anal 136(2):191–200MathSciNetCrossRefGoogle Scholar
- 14.Ciarlet PG, Lods V, Miara B (1996) Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations. Arch Ration Mech Anal 136(2):163–190MathSciNetCrossRefGoogle Scholar
- 15.Dauge M, Faou E, Yosibash Z (2004) Plates and shells: asymptotic expansions and hierarchic models. In: Stein E, de Borst R, Hughes T (eds) Encyclopedia of computational mechanics, chap 8, vol 2. Wiley Online Library, New York, pp 199–236Google Scholar
- 16.Demlow A (2009) Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J Numer Anal 47(2):805–827MathSciNetCrossRefGoogle Scholar
- 17.Dziuk G, Elliott C (2013) Finite element methods for surface PDEs. Acta Numer 22:289–396MathSciNetCrossRefGoogle Scholar
- 18.Echter R, Oesterle B, Bischoff M (2013) A hierarchic family of isogeometric shell finite elements. Comput Methods Appl Mech Eng 254:170–180MathSciNetCrossRefGoogle Scholar
- 19.Fike JA, Alonso JJ (2011) The development of hyper-dual numbers for exact second-derivative calculations. In: 49th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, Orlando, FloridaGoogle Scholar
- 20.Fries TP, Schöllhammer D (2017) Higher-order meshing of implicit geometries part II: approximations on manifolds. Comput Methods Appl Mech Eng 326:270–297MathSciNetCrossRefGoogle Scholar
- 21.Gfrerer MH, Schanz M (2017) Code verification examples based on the method of manufactured solutions for Kirchhoff–Love and Reissner–Mindlin shell analysis. Eng Comput. https://doi.org/10.1007/s00366-017-0572-4 CrossRefGoogle Scholar
- 22.Gfrerer MH, Schanz M (2018) A high-order FEM with exact geometry description for the Laplacian on implicitly defined surfaces. Int J Numer Methods Eng. https://doi.org/10.1002/nme.5779 MathSciNetCrossRefGoogle Scholar
- 23.Hosseini S, Remmers JJ, Verhoosel CV, Borst R (2013) An isogeometric solid-like shell element for nonlinear analysis. Int J Numer Methods Eng 95(3):238–256MathSciNetCrossRefGoogle Scholar
- 24.Kiendl J, Bletzinger KU, Linhard J, Wüchner R (2009) Isogeometric shell analysis with Kirchhoff–Love elements. Comput Methods Appl Mech Eng 198(49):3902–3914MathSciNetCrossRefGoogle Scholar
- 25.Lehrenfeld C (2016) High order unfitted finite element methods on level set domains using isoparametric mappings. Comput Methods Appl Mech Eng 300:716–733MathSciNetCrossRefGoogle Scholar
- 26.Lorensen W, Cline H (1987) Marching cubes: a high resolution 3D surface construction algorithm. In: Proceedings of the 14th annual conference on computer graphics and interactive techniques, ACM, New York, NY, USA, SIGGRAPH’87, pp 163–169Google Scholar
- 27.Macneal RH, Harder RL (1985) A proposed standard set of problems to test finite element accuracy. Finite Elem Anal Des 1(1):3–20CrossRefGoogle Scholar
- 28.Oesterle B, Sachse R, Ramm E, Bischoff M (2017) Hierarchic isogeometric large rotation shell elements including linearized transverse shear parametrization. Comput Methods Appl Mech Eng 321:383–405MathSciNetCrossRefGoogle Scholar
- 29.Pimenta PM, Campello EMB (2009) Shell curvature as an initial deformation: a geometrically exact finite element approach. Int J Numer Methods Eng 78(9):1094–1112MathSciNetCrossRefGoogle Scholar
- 30.Ramm E, Wall W (2004) Shell structures—a sensitive interrelation between physics and numerics. Int J Numer Methods Eng 60(1):381–427MathSciNetCrossRefGoogle Scholar
- 31.Rank E, Düster A, Nübel V, Preusch K, Bruhns O (2005) High order finite elements for shells. Comput Methods Appl Mech Eng 194(21):2494–2512CrossRefGoogle Scholar
- 32.Schöberl J, Zaglmayr S (2005) High order Nédélec elements with local complete sequence properties. Compel 24(2):374–384MathSciNetCrossRefGoogle Scholar
- 33.Takacs T, Jüttler B (2011) Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis. Comput Methods Appl Mech Eng 200(49):3568–3582MathSciNetCrossRefGoogle Scholar
- 34.Weingarten J (1861) Ueber eine Klasse auf einander abwickelbarer Flächen. J Reine Angew Math 59:382–393MathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.