# Kirchhoff–Love shell theory based on tangential differential calculus

- 129 Downloads

## Abstract

The Kirchhoff–Love shell theory is recasted in the frame of the tangential differential calculus (TDC) where differential operators on surfaces are formulated based on global, three-dimensional coordinates. As a consequence, there is no need for a parametrization of the shell geometry implying curvilinear surface coordinates as used in the classical shell theory. Therefore, the proposed TDC-based formulation also applies to shell geometries which are zero-isosurfaces as in the level-set method where no parametrization is available in general. For the discretization, the TDC-based formulation may be used based on surface meshes implying element-wise parametrizations. Then, the results are equivalent to those obtained based on the classical theory. However, it may also be used in recent finite element approaches as the TraceFEM and CutFEM where shape functions are generated on a background mesh without any need for a parametrization. Numerical results presented herein are achieved with isogeometric analysis for classical and new benchmark tests. Higher-order convergence rates in the residual errors are achieved when the physical fields are sufficiently smooth.

## Keywords

Shells Tangential differential calculus TDC Isogeometric analysis IGA Manifolds## 1 Introduction

*local*coordinates living on the manifold or on

*global*coordinates of the surrounding, three-dimensional space.

In the first case, the curved surface is parametrized by two parameters, i.e., there is a given map from the two-dimensional parameter space to the three-dimensional physical space, see Fig. 1a. For the definition of geometrical quantities and surface operators, co- and contra-variant base vectors and Christoffel-symbols naturally occur. It is important to note that a parametrization of a surface is not unique, hence, there are infinitely many maps which result in the same curved surface. Obviously, the physical modeling must be independent of a concrete parametrization, which suggests the existence of a parametrization-free formulation.

In the second case, the geometric quantities and surface operators are based on global coordinates as done in the tangential differential calculus (TDC) [15, 25, 28]. Then, a model may also be defined even if a parametrization of a curved surface does not exist, for example, when it is a zero-isosurface of a scalar function in three dimensions following the level-set method [21, 22, 39, 43]. When the physical modeling is based on the TDC, i.e., on global coordinates, it is applicable to surfaces which are parametrized or not. In this sense, the TDC-based approach is more general than approaches based on local coordinates. Models based on the TDC are found in various applications, see [16, 17, 18, 22] for scalar problems such as heat flow and [20, 31] for flow problems on manifolds. In the context of structure mechanics, this approach is used in [29] for curved beams, in [25, 26, 28] for membranes, and in [27] for flat shells embedded in \(\mathbb {R}^3\).

Herein, we apply the TDC for the reformulation of the classical Kirchhoff–Love shell theory which is typically formulated based on a given parametrization. Based on the TDC, it is possible to also formulate the boundary value problem (BVP) for shell geometries where no parametrization is given as for the example in Fig. 1b: the cupola with radius r is given by the zero-isosurface of \(\phi (\varvec{x}) = \Vert \varvec{x}\Vert -r\) with \(\varvec{x} \in [-r,r]^2 \times [0,r]\) and the mechanical response to the force *F* is sought. As mentioned before, the TDC-based formulation is also valid when a parametrization is available; it is then equivalent to the classical formulation based on local coordinates.

Other attempts to parametrization-free formulations of the Kirchhoff–Love shell theory are found, e.g., in [11, 12, 13, 14] with a mathematical focus and in [33, 47, 50] from an engineering perspective, however, only with focus on displacements. Herein, the Kirchhoff–Love shell theory is recasted in the frame of the TDC including all relevant mechanical aspects. For the first time, the parametrization-free strong form of the Kirchhoff–Love shell is given and taken as the starting point to derive the weak form. Then, boundary terms for the relevant boundary conditions of Kirchhoff–Love shell theory are naturally achieved. Furthermore, mechanical quantities such as moments, normal and shear forces are defined based on global coordinates and it is shown how (parametrization-)invariant quantities such as principal moments are computed. Finally, the strong form of Kirchhoff–Love shells is also found highly useful to define residual errors in the numerical results. Of course, evaluating this error in the strong form requires up to forth-order derivatives on the surface, which is implementationally quite some effort. The advantage, however, is that one may then confirm higher-order convergence rates in the corresponding error norm for suitable shell test cases. This is, otherwise, very difficult as exact solutions for shells are hardly available and classical benchmark tests typically give only selected scalar quantities, often with moderate accuracy.

For the numerical solution of shells, i.e., the approximation of the shell BVP based on numerical methods, we distinguish two fundamentally different approaches. The first is a classical finite element analysis based on a surface mesh, labelled Surface FEM herein [16, 18, 20, 22]. Once a surface mesh is generated, it implies element-wise parametrizations for the shell geometry, see Fig. 1c, no matter whether the underlying (analytic) geometry was parametrized or implied by level sets. In this case, classical shell theory based on parametrizations is suitable at least for the discretized geometry. The proposed TDC-based formulation is suitable as well which shall be seen in the numerical results. The other numerical approach is to use a three-dimensional background mesh into which the curved shell surface is embedded, cf. Fig. 1d. Then, the shape functions of the (three-dimensional) background elements are only evaluated on the shell surface and no parametrization (and surface mesh) is needed to furnish basis functions for the approximation. For these methods, e.g., labelled CutFEM [6, 7, 8, 19] or TraceFEM [23, 37, 38, 42], applied to the case of shell mechanics, it is no longer possible to rely on classical parametrization-based formulations of the shell mechanics, however, the proposed TDC-based formulation is still applicable.

For the numerical results presented herein, the continuous weak form of the BVP is discretized with the Surface FEM [16, 18, 20, 22] using NURBS as trial and test functions as proposed by Hughes et al. [10, 30] due to the continuity requirements of Kirchhoff–Love shells. The boundary conditions are weakly enforced via Lagrange multiplies [51]. The situation is similar to [2, 32, 35, 36], however, based on the proposed view point, the implementation is quite different. In particular, when PDEs on manifolds from other application fields than shell mechanics are also of interest (e.g., when transport problems [16, 17, 18] or flow problems [20, 31] on curved surfaces are considered), there is a unified and elegant way to handle this by computing surface gradients applied to finite element shape functions which simplifies the situation considerably. In that sense one may shift significant parts of the implementation needed for shells to the underlying finite element technology and recycle this in other situations where PDEs on surfaces are considered.

We summarize the advantages of the TDC-based formulation of Kirchhoff–Love shells: (1) the definition of the BVP does not need a parametrization of the surface (though it can also handle the classical situation where a parametrization is given), (2) the TDC-based formulation is also suitable for very recent finite element technologies such as CutFEM and TraceFEM (though the typical approach based on the Surface FEM or IGA is also possible and demonstrated herein), (3) the implementation is advantagous in finite element (FE) codes where other PDEs on manifolds are considered as well due to the split of FE technology and application. From a didactic point of view, it may also be advantageous that troubles with curvilinear coordinates (co- and contra-variance, Christoffel-symbols) are avoided in the TDC-based approach where surface operators and geometric quantities are expressed in tensor notation.

The outline of the paper is as follows: In Sect. 2, important surface quantities are defined, and an introduction to the tangential differential calculus (TDC) is given. In Sect. 3, the classical linear Kirchhoff–Love shell equations under static loading are recast in terms of the TDC. Stress resultants such as membrane forces, bending moments, transverse shear forces and corner forces are defined. In Sect. 4, implementational aspects are considered. The element stiffness matrix and the resulting system of linear equations are shown. The implementation of boundary conditions based on Lagrange multipliers is outlined. Finally, in Sect. 5, numerical results are presented. The first example is a flat shell embedded in \(\mathbb {R}^3\), where an analytical solution is available. The second and third example are parts of popular benchmarks as proposed in [2]. In the last example, a more general geometry without analytical solution or reference displacement is considered. The error is measured in the strong form of the equilibrium in order to verify the proposed approach and higher-order convergence rates are achieved.

## 2 Preliminaries

The definition of the normal vector depends on whether the shell geometry is based on a parametrization or not. In the first case (cf. Fig. 2a), the shell geometry results from a map \(\varvec{x}(\varvec{r})\). Then, the normal vector \(\varvec{n}_{\varGamma }\) of the shell surface is determined by a cross-product of the columns of the Jacobi matrix \(\mathbf J (\varvec{r})=\nicefrac { \partial \varvec{x}}{ \partial \varvec{r}}\). The resulting geometric quantities, surface operators, and models in this case are parametrization-based.

In the case where the shell geometry is implied by the zero-isosurface of a level-set function \(\phi (\varvec{x})\) (cf. Fig. 2b) and no parametrization is available, the normal vector may be determined by \(\varvec{n}_{\varGamma }= \nicefrac {\nabla \phi }{\Vert \nabla \phi \Vert }\). All resulting quantities including the BVP of the Kirchhoff–Love shell are parametrization-free in this case. Of course, when in the wake of discretizing the BVP, the Surface FEM is used for the approximation, then a surface mesh of the shell geometry is needed and the surface elements do imply a parametrization again. It was already mentioned above, that other numerical methods such as the TraceFEM and CutFEM do not rely on a surface mesh. In this case, the countinuous and discrete BVP for the shell are truly parametrization-free.

In addition to the normal vector on the surface, along the boundary \( \partial {\varGamma }\) there is an associated tangential vector \(\varvec{t}_{ \partial {\varGamma }} \in \mathbb {R}^3\) pointing in the direction of \( \partial {\varGamma }\) and a co-normal vector \(\varvec{n}_{ \partial {\varGamma }} = \varvec{n}_{\varGamma }\times \varvec{t}_{ \partial {\varGamma }} \in \mathbb {R}^3\) pointing “outwards” and being perpendicular to the boundary yet in the tangent plane of the surface \({\varGamma }\). For the proof of equivalence of both cases we refer to, e.g., [18].

### 2.1 Tangential differential calculus

The TDC provides a framework to define differential operators avoiding the use of classical differential geometric methods based on local coordinate systems and Christoffel symbols. In the following, an overview of the operators and relations in the frame of the TDC are presented. For simplicity, we restrict ourselves to the case of surfaces embedded in the three dimensional space. However, the shown relations and definitions may be adopted to other situations accordingly (e.g., curved lines embedded in 2D or 3D). An introduction from a more mathematical point of view is given in [15, 25, 31].

#### 2.1.1 Orthogonal projection operator \(\mathbf P \)

#### 2.1.2 Tangential gradient of scalar functions

*u*in a neighbourhood \(\mathcal {U}\) of the manifold \({\varGamma }\). Alternatively, \(\tilde{u}\) is given as a function in global coordinates \(\tilde{u}(\varvec{x}) : \mathbb {R}^3 \rightarrow \mathbb {R}\) and only evaluated at the manifold \(\tilde{u}_{|_{\varGamma }} = u\).

#### 2.1.3 Tangential gradient of vector-valued functions

*directional*gradient of \(\varvec{v}\) defined as

*not*in the tangent space of the surface, in general. A projection of the directional gradient to the tangent space leads to the

*covariant*gradient of \(\varvec{v}\) and is defined as

In the following, partial surface derivatives of scalar functions are denoted as \( \partial ^{\varGamma }_{x_i} u\) or \(u_{,i}^{\varGamma }\) with \(i = 1,\,2,\,3\). Partial surface derivatives of vector or tensor components are denoted as \(v_{i,j}^{\text {dir} }\) for directional and \(v_{i,j}^{\text {cov} }\) for covariant derivatives with \(i,j = 1,\,2,\,3\).

#### 2.1.4 Tangential gradient of tensor functions

#### 2.1.5 Second-order tangential derivatives

*u*is defined by

#### 2.1.6 Tangential divergence operators

*and*\(\mathbf A \) is an in-plane tensor.

#### 2.1.7 Weingarten map and curvature

The Gauß curvature is defined as the product of the principal curvatures \(K = \prod _{i=1}^{2}\kappa _i\) and the mean curvature is introduced as \(\varkappa = \kappa _1 + \kappa _2 = \text {tr} (\mathbf H )\).

#### 2.1.8 Divergence theorems in terms of tangential operators

## 3 The shell equations

With these assumptions, an analytical pre-integration with respect to the thickness leads to stress resultants such as normal forces and bending moments. The equilibrium in strong form is then expressed in terms of the stress resultants. Finally, the transverse shear forces may be identified via equilibrium considerations.

### 3.1 Kinematics

*t*is defined by

### 3.2 Constitutive equation

#### 3.2.1 Stress resultants

The stress tensor is only a function of the middle surface displacement vector \(\varvec{u}\), the difference vector \(\varvec{w}(\varvec{u})\) and the thickness parameter \(\zeta \). This enables an analytical pre-integration with respect to the thickness and stress resultants can be identified. The following quantities are equivalent to the stress resultants in the classical theory [1, 45], but they are expressed in terms of the TDC using a global Cartesian coordinate system.

*not*the physical normal force tensor. This tensor only appears in the variational formulation, see Sect. 4. The physical normal force tensor \(\mathbf n ^{\text {real} }_{\varGamma }\) is defined by

### 3.3 Equilibrium

*flat*Kirchhoff–Love structures embedded in \(\mathbb {R}^3\) the divergence of an in-plane tensor is a tangential vector, as already mentioned in Sect. 2.1. Therefore, the definition of the transverse shear force vector in [27] is in agreement with the obtained transverse shear force vector herein.

#### 3.3.1 Equilibrium in weak form

#### 3.3.2 Boundary conditions

As well known in the classical Kirchhoff–Love shell theory, special attention needs to be paid to the boundary conditions. In the following, the boundary terms of the weak form in Eq. (38) are rearranged in order to derive the effective boundary forces.

*C*. The new boundary term are the Kirchhoff forces or corner forces. Note that if the boundary of the shell is smooth, the corner forces vanish. Finally, the integral over the Neumann boundary in Eq. (38) is expressed in terms of the well-known effective boundary forces and the bending moment along the boundary

Set of common boundary conditions

Clamped edge | \(u_{\varvec{t}_{ \partial {\varGamma }}} = 0\) | \(u_{\varvec{n}_{ \partial {\varGamma }}} = 0\) | \(u_{\varvec{n}_{\varGamma }} = 0\) | \(\omega _{\varvec{t}_{ \partial {\varGamma }}} = 0\) |

Simply supported edge | \(u_{\varvec{t}_{ \partial {\varGamma }}} = 0\) | \(u_{\varvec{n}_{ \partial {\varGamma }}} = 0\) | \(u_{\varvec{n}_{\varGamma }} = 0\) | \(m_{\varvec{t}_{ \partial {\varGamma }}} = 0\) |

Symmetry support | \(\tilde{p}_{\varvec{t}_{ \partial {\varGamma }}} = 0\) | \(u_{\varvec{n}_{ \partial {\varGamma }}} = 0\) | \(\tilde{p}_{\varvec{n}_{\varGamma }} = 0\) | \(\omega _{\varvec{t}_{ \partial {\varGamma }}} = 0\) |

Free edge | \(\tilde{p}_{\varvec{t}_{ \partial {\varGamma }}} = 0\) | \(\tilde{p}_{\varvec{n}_{ \partial {\varGamma }}} = 0\) | \(\tilde{p}_{\varvec{n}_{\varGamma }} = 0\) | \(m_{\varvec{t}_{ \partial {\varGamma }}} = 0\) |

## 4 Implementational aspects

The continuous weak form is discretized using isogeometric analysis as proposed by Hughes et al. [10, 30]. The NURBS patch *T* is the middle surface of the shell and the elements \(\tau _i\ ( i=1,\,\ldots ,\,n_{\text {Elem} })\) are defined by the knot spans of the patch. The mesh is then defined by the union of the elements \({\varGamma }= \bigcup \limits _{\tau \in T} \tau \).

There is a fixed set of local basis functions \(\lbrace N_i^k(\varvec{r})\rbrace \) of order *k* with \(i = 1,\,\ldots ,\,n_k\) being the number of control points and the displacements \(\lbrace \hat{u}_i,\,\hat{v}_i,\,\hat{w}_i\rbrace \) stored at the control points *i* are the degrees of freedom. Using the isoparametric concept, the shape functions \(N_i^k(\varvec{r})\) are NURBS of order *k*. The surface derivatives of the shape functions are computed as defined in Sect. 2 , similar as in the Surface FEM [16, 18, 20, 22] using NURBS instead of Lagrange polynomials as ansatz and test functions. The shape functions of order \(k \ge 2\) are in the function space \(\mathcal {V}\), see Eq. (39). In fact, the used shape functions are in the Sobolev space \(\mathcal {H}^k({\varGamma })^3 \subset \mathcal {V}\) iff \(k \ge 2\).

*a*and

*b*. The matrix \(\hat{\mathbf{K }}\) is determined by directional first-order derivatives of the shape functions \(\varvec{N}\). One may recognize that the structure of the matrix \(\hat{\mathbf{K }}\) is similar to the stiffness matrix of 3D linear elasticity problems. For the bending part we have

In this sense, a significant part of the complexity of implementing shells is shifted to finite element technology and may be recycled for any kind of boundary value problems on curved surfaces in \(\mathbb {R}^3\). Examples are transport problems [16, 17, 18] or flow problems [20, 31] on curved surfaces. We expect that future implementations in finite element software will provide frameworks for solving PDEs on manifolds and, based, e.g., on this work will also apply to shells. In order to emphasize the differences in the implementation, example Matlab®-codes for the proposed TDC-based formulation and the classical parametrization-based formulation are given in Sect. A, clearly highlighting the differences.

*i*is defined as

## 5 Numerical results

The numerical results are achieved using NURBS functions for the geometry and shape function definition, following the methodology of isogeometric analysis (IGA) [3, 24, 30, 32, 34]. The definition of NURBS is omitted here for brevity but is found at numerous references in the frame of IGA, e.g., [10, 40].

The obtained shell equations are carefully verified and compared to the classical approach with different test cases. As already mentioned above the proposed approach leads to an equivalent stiffness matrix for arbitrary curved and non-curved shells. Consequently, the same convergence properties as shown, e.g., in [9, 32] are expected. In the following, the results of the convergence analyses of a flat shell embedded in \(\mathbb {R}^3\), the Scordelis-Lo roof, and the pinched cylinder test (part of the shell obstacle course proposed by Belytschko et al. [2]) are shown. Furthermore, a new test case with a challenging geometry is proposed which features smooth solutions enabling higher-order convergence rates. These rates are confirmed in the residual error as no analytic solution exists, see Sect. 5.4. Other examples (e.g., pinched hemispherical shell, shells of revolution, etc.) have been considered but are omitted here for brevity.

In the convergence studies, NURBS patches with different orders and numbers of knot spans in each direction are employed. This is equivalent to meshes with higher-order elements and \(n=\lbrace 2,\,4,\,8,\,16,\,32\rbrace \) elements per side are used. The orders are varied as \(p=\lbrace 2,\,3,\,4,\,5,\,6\rbrace \).

### 5.1 Flat shell embedded in \(\mathbb {R}^3\)

In Fig. 7, the solution of the shell is illustrated. The displacements are scaled by two orders of magnitude. The colours on the deformed surface indicate the Euclidean norm of the displacement field \(\Vert \varvec{u} \Vert \).

The results of the convergence analysis are shown in Fig. 8. The curves are plotted as a function of the element size \(\nicefrac {1}{n}\) (which is rather a characteristic length of the knot spans). The dotted lines indicate the theoretical optimal order of convergence. In Fig. 8a, the relative \(L_2\)-error of the primal variable (displacements) is shown. Optimal higher-order convergence rates \(\mathcal {O}(p+1)\) are achieved. In the figures Fig. 8b–d, the relative \(L_2\)-errors of the normal forces (membrane forces), bending moments and transverse shear forces are plotted. For all stress resultants the theoretical optimal orders of convergence are achieved. It is clear that the same results were obtained if the results are computed for the purely two-dimensional case as, e.g., in [9].

### 5.2 Scordelis-Lo roof

In Fig. 10b, the convergence of the maximum displacement \(u_{z,\max }\) is plotted up to polynomial order of \(p=6\) as a function of the element (knot span) size. It is clearly seen that the expected results are achieved, with increasing accuracy for higher-order NURBS. Due to the lack of a more accurate reference solutions, it is not useful to show these results in a double-logarithmic diagram as usual for error plots. The style of presentation follows those of many other references such as, e.g., in [2, 9, 32].

### 5.3 Pinched cylinder

As in the example before, in Fig. 12b, the convergence to a normalized reference displacement as a function of the element size is plotted. The results converge with the expected behaviour as shown in [9, 32]. It is noted that due to the singularity in some mechanical quantities due to the single force, higher-order convergence rates are not possible here. However, the improvement for increasing the order of the NURBS is still seen in the figure. An additional grading of the elements in order to better resolve the singularity would have further improved the situation but is omitted here.

### 5.4 Flower shaped shell

In Fig. 14a, the deformed shell is illustrated. The displacement field is scaled by one order of magnitude. In Fig. 14b, the results of the convergence analysis are plotted. Due to the fact that fourth-order derivatives need to be computed, at least fourth-order shape functions are required. The theoretical optimal order of convergence is \(\mathcal {O}(p-3)\) if the solution is smooth enough. One may observe that higher-order convergence rates are achieved, however, rounding-off errors and the conditioning may slightly influence the convergence. Nevertheless, the results are excellent also given the fact that higher-order accurate results for shells (given in double-logarithmic error plots) are the exception.

## 6 Conclusions and outlook

The linear Kirchhoff–Love shell theory is reformulated in terms of the TDC using a global Cartesian coordinate system and tensor notation. The resulting model equations apply to shell geometries which are parametrized or not. For example, a parametrization may not be available when shell geometries are implied by the level-set method. Because the TDC-based formulation holds in both cases, it may be seen as a generalization to the classical shell theory which is based on parametrizations and curvilinear coordinates.

The TDC-based strong form is used as the starting point to consistently obtain the weak form including all boundary terms well-known in the Kirchhoff–Love theory. Mechanical stress-resultants such as moments, normal and shear forces are defined in global coordinates. Furthermore, the strong form may be used in the numerical results to compute residual errors and thus enable convergence analyses even without the knowledge of exact solutions which, for shells, are scarce.

For the discretization, the Surface FEM is used with NURBS as trial and test functions. That is, an isogeometric approach is chosen due to continuity requirements. In this case, the presence of a surface mesh (i.e., a NURBS patch), implies a parametrization and although the involved equations and the resulting implementations vary significantly, it is seen that the classical, parametrization-based and the proposed TDC-based formulation are equivalent. For a generic finite element framework enabling various implementations for PDEs on manifolds (in addition to only shells), the TDC-based approach is benefitial, because surface gradients of shape functions may be computed beforehand and are independent of the application.

The numerical results confirm higher-order convergence rates. As mentioned, based on the residual errors, a framework for the verification of complex test cases is presented. There is a large potential in the parametrization-free reformulation of shell models, because the obtained PDEs may be discretized with new finite element techniques such as TraceFEM or CutFEM based on implicitly defined surfaces. In this case, neither the problem statement nor the discretization is based on a parametrization.

## Notes

### Acknowledgements

Open access funding provided by Graz University of Technology.

## Supplementary material

## References

- 1.Başar Y, Krätzig WB (1985) Mechanik der Flächentragwerke. Vieweg+Teubner Verlag, BraunschweigCrossRefGoogle Scholar
- 2.Belytschko T, Stolarski H, Liu WK, Carpenter N, Ong JSJ (1985) Stress projection for membrane and shear locking in shell finite elements. Comput Methods Appl Mech Eng 51:221–258MathSciNetCrossRefGoogle Scholar
- 3.Benson DJ, Bazilevs Y, Hsu MC, Hughes TJR (2010) Isogeometric shell analysis: the Reissner–Mindlin shell. Comput Methods Appl Mech Eng 199:276–289MathSciNetCrossRefGoogle Scholar
- 4.Bischoff M, Bletzinger KU, Wall WA, Ramm E (2004) Models and finite elements for thin-walledstructures, chapter 3. In: Encyclopedia of computational mechanics. Wiley, ChichesterGoogle Scholar
- 5.Blaauwendraad J, Hoefakker JH (2014) Structural shell analysis. Solid mechanics and its applications, vol 200. Springer, BerlinCrossRefGoogle Scholar
- 6.Burman E, Claus S, Hansbo P, Larson MG, Massing A (2015) CutFEM: discretizing geometry and partial differential equations. Int J Numer Methods Eng 104:472–501MathSciNetCrossRefGoogle Scholar
- 7.Burman E, Elfverson D, Hansbo P, Larson MG, Larsson K (2018) Shape optimization using the cut finite element method. Comput Methods Appl Mech Eng 328:242–261MathSciNetCrossRefGoogle Scholar
- 8.Cenanovic M, Hansbo P, Larson MG (2016) Cut finite element modeling of linear membranes. Comput Methods Appl Mech Eng 310:98–111MathSciNetCrossRefGoogle Scholar
- 9.Cirak F, Ortiz M, Schröder P (2000) Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. Int J Numer Methods Eng 47(12):2039–2072CrossRefGoogle Scholar
- 10.Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, ChichesterCrossRefGoogle Scholar
- 11.Delfour MC, Zolésio JP (1994) Shape analysis via oriented distance functions. J Funct Anal 123:129–201MathSciNetCrossRefGoogle Scholar
- 12.Delfour MC, Zolésio JP (1995) A boundary differential equation for thin shells. J Differ Equ 119:426–449MathSciNetCrossRefGoogle Scholar
- 13.Delfour MC, Zolésio JP (1996) Tangential differential equations for dynamical thin shallow shells. J Differ Equ 128:125–167MathSciNetCrossRefGoogle Scholar
- 14.Delfour MC, Zolésio JP (1997) Differential equations for linear shells comparison between intrinsic and classical. In: Advances in mathematical sciences: CRM’s 25 years (Montreal, PQ, 1994), Vol. 11 of CRM proceedings and lecture notes, Providence, Rhode IslandGoogle Scholar
- 15.Delfour MC, Zolésio JP (2011) Shapes and geometries: metrics, analysis, differential calculus, and optimization. SIAM, PhiladelphiaCrossRefGoogle Scholar
- 16.Demlow A (2009) Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J Numer Anal 47:805–827MathSciNetCrossRefGoogle Scholar
- 17.Dziuk G (1988) Finite elements for the beltrami operator on arbitrary surfaces: chapter 6. Springer, Berlin, pp 142–155Google Scholar
- 18.Dziuk G, Elliott CM (2013) Finite element methods for surface PDEs. Acta Numer 22:289–396MathSciNetCrossRefGoogle Scholar
- 19.Elfverson D, Larson MG, Larsson K (2018) A new least squares stabilized Nitsche method for cut isogeometric analysis. ArXiv e-prints ArXiv:1804.05654
- 20.Fries TP (2018) Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds. Int J Numer Methods Fluids 88:55–78. https://doi.org/10.1002/fld.4510 MathSciNetCrossRefGoogle Scholar
- 21.Fries TP, Omerović S, Schöllhammer D, Steidl J (2017) Higher-order meshing of implicit geometries—part I: integration and interpolation in cut elements. Comput Methods Appl Mech Eng 313:759–784MathSciNetCrossRefGoogle Scholar
- 22.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
- 23.Grande J, Reusken A (2016) A higher order finite element method for partial differential equations on surfaces. SIAM 54:388–414MathSciNetzbMATHGoogle Scholar
- 24.Guo Y, Ruess M, Schillinger D (2017) A parameter-free variational coupling approach for trimmed isogeometric thin shells. Comput Mech 59:693–715MathSciNetCrossRefGoogle Scholar
- 25.Gurtin ME, Murdoch IA (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57:291–323MathSciNetCrossRefGoogle Scholar
- 26.Hansbo P, Larson MG (2014) Finite element modeling of a linear membrane shell problem using tangential differential calculus. Comput Methods Appl Mech Eng 270:1–14MathSciNetCrossRefGoogle Scholar
- 27.Hansbo P, Larson MG (2017) Continuous/discontinuous finite element modelling of Kirchhoff plate structures in \(\mathbb{R}^{3}\) using tangential differential calculus. Comput Mech 60:693–702MathSciNetCrossRefGoogle Scholar
- 28.Hansbo P, Larson MG, Larsson F (2015) Tangential differential calculus and the finite element modeling of a large deformation elastic membrane problem. Comput Mech 56:87–95MathSciNetCrossRefGoogle Scholar
- 29.Hansbo P, Larson MG, Larsson K (2014) Variational formulation of curved beams in global coordinates. Comput Mech 53:611–623MathSciNetCrossRefGoogle Scholar
- 30.Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195MathSciNetCrossRefGoogle Scholar
- 31.Jankuhn T, Olshanskii MA, Reusken A (2017) Incompressible fluid problems on embedded surfaces modeling and variational and formulations. ArXiv e-prints arXiv:1702.02989
- 32.Kiendl J, Bletzinger K-U, Linhard J, Wüchner R (2009) Isogeometric shell analysis with Kirchhoff–Love elements. Comput Methods Appl Mech Eng 198:3902–3914MathSciNetCrossRefGoogle Scholar
- 33.Lebiedzik C (2007) Exact boundary controllability of a shallow intrinsic shell model. J Math Anal Appl 335:584–614MathSciNetCrossRefGoogle Scholar
- 34.Nguyen VP, Anitescu C, Bordas SPA, Rabczuk T (2015) Isogeometric analysis: an overview and computer implementation aspects. Math Comput Simul 117:89–116MathSciNetCrossRefGoogle Scholar
- 35.Nguyen-Thanh N, Valizadeh N, Nguyen MN, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, Lorenzis L, De Rabczuk T (2015) An extended isogeometric thin shell analysis based on Kirchhoff–Love theory. Comput Methods Appl Mech Eng 284:265–291MathSciNetCrossRefGoogle Scholar
- 36.Nguyen-Thanh N, Zhou K, Zhuang X, Areias P, Nguyen-Xuan H, Bazilevs Y, Rabczuk T (2017) Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Comput Methods Appl Mech Eng 316:1157–1178MathSciNetCrossRefGoogle Scholar
- 37.Olshanskii MA, Reusken A (2017) Trace finite element methods for PDEs on surfaces. In: Lecture notes in computational science and engineering, vol 121, pp 211–258Google Scholar
- 38.Olshanskii MA, Xu X (2017) A trace finite element method for PDEs on evolving surfaces. SIAM 39:A1301–A1319MathSciNetzbMATHGoogle Scholar
- 39.Osher S, Fedkiw RP (2003) Level set methods and dynamic implicit surfaces. Springer, BerlinCrossRefGoogle Scholar
- 40.Piegl L, Tiller W (1997) The NURBS book (monographs in visual communication), 2nd edn. Springer, BerlinCrossRefGoogle Scholar
- 41.Radwańska M, Stankiewicz A, Wosatko A, Pamin J (2017) Plate and shell structures. Wiley, ChichesterCrossRefGoogle Scholar
- 42.Reusken A (2014) Analysis of trace finite element methods for surface partial differential equations. IMA J Numer Anal 35:1568–1590MathSciNetCrossRefGoogle Scholar
- 43.Sethian JA (1999) Level set methods and fast marching methods, 2nd edn. Cambridge University Press, CambridgezbMATHGoogle Scholar
- 44.Simo JC, Fox DD (1989) On a stress resultant geometrically exact shell model—part I: formulation and optimal parametrization. Comput Methods Appl Mech Eng 72:267–304CrossRefGoogle Scholar
- 45.Simo JC, Fox DD, Rifai MS (1989) On a stress resultant geometrically exact shell model—part II: the linear theory; computational aspects. Comput Methods Appl Mech Eng 73:53–92CrossRefGoogle Scholar
- 46.Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells, 2nd edn. McGraw-Hill Book Company Inc, New YorkzbMATHGoogle Scholar
- 47.van Opstal TM, van Brummelen EH, van Zwieten GJ (2015) A finite-element/boundary-element method for three-dimensional, large-displacement fluid structure-interaction. Comput Methods Appl Mech Eng 284:637–663MathSciNetCrossRefGoogle Scholar
- 48.Walker SW (2015) The shapes of things: a practical guide to differential geometry and the shape derivative. Advances in design and control. SIAM, PhiladelphiaCrossRefGoogle Scholar
- 49.Wempner G, Talaslidis D (2002) Mechanics of solids and shells: theories and approximations. CRC Press, Boca RatonCrossRefGoogle Scholar
- 50.Yao PF (2009) On shallow shell equations. Discret Contin Dyn Syst Ser S 2:697–722MathSciNetCrossRefGoogle Scholar
- 51.Zienkiewicz O, Taylor R, Zhu JZ (2013) The finite element method: its basis and fundamentals, 7th edn. Elsevier, OxfordzbMATHGoogle 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.