Isogeometric B-Rep analysis (IBRA) firstly introduced in [20] allows for performing *Isogeometric Analysis* (IGA) on real-world CAD models which involve trimmed multipatches. In this section the isogeometric analysis of lightweight structures on trimmed NURBS multipatches is briefly presented. Accordingly, membranes and thin shells of Kirchhoff-Love shell type are used, see in [38] for more information.

### Differential geometry of surfaces

Herein, a brief introduction to the differential geometry of surfaces is provided and the underlying notions are used in the sequel, see in [39] for more information. Given is a surface \(\Omega \subset {\mathbb {R}}^3\) with parametric image \({\hat{\Omega }} \subset {\mathbb {R}}^2\). Given also a parametrization of that surface \(\mathbf {S}:{\hat{\Omega }} \rightarrow \Omega \) which is well-defined *almost everywhere* (a.e.), that is, every parametric location \((\theta _1 , \theta _2) \in {\hat{\Omega }}\) is mapped onto a unique Cartesian location \(\mathbf {X} = (X_1,X_2,X_3)\) through map \(\mathbf {S}\) a.e. in \({\hat{\Omega }}\) (see “Non-uniform rational b-spline surfaces” section for the NURBS parametrization of a surface). Accordingly, a *covariant* basis may be constructed as follows,

$$\begin{aligned} \mathbf {A}_{\alpha }&= \mathbf {S}_{,\alpha } \;, \end{aligned}$$

(8a)

$$\begin{aligned} \mathbf {A}_3&= \frac{1}{{\bar{j}}} \mathbf {A}_1 \times \mathbf {A}_2 \;, \end{aligned}$$

(8b)

where \((\bullet )_{,\alpha } = \partial (\bullet )/\partial \theta _{\alpha }\) and \({\bar{j}} = \Vert \mathbf {A}_1 \times \mathbf {A}_2 \Vert _2\). Map \(\mathbf {S}\) is then well-defined at parametric locations where \({\bar{j}} \ne 0\). The components of the *metric* tensor \(\mathbf {A} = A_{\alpha \beta } \, \mathbf {A}_{\alpha } \otimes \mathbf {A}_{\beta }\) (also known as the *first fundamental form* of a surface) are given by,

$$\begin{aligned} A_{\alpha \beta } = \mathbf {A}_{\alpha } \cdot \mathbf {A}_{\beta } \;. \end{aligned}$$

(9)

The *contravariant* components of the metric coefficient tensor, namely, \(A^{\beta \gamma }\) can be obtained by the relation \(A_{\alpha \beta } A^{\beta \gamma } = \delta _{\alpha }^{\;\gamma }\), where \(\delta _{\alpha }^{\;\gamma }\) stands for the *Kronecker* delta symbol, that is, \(\delta _{\alpha }^{\;\gamma } = 1\) for \(\alpha = \gamma \) and \(\delta _{\alpha }^{\;\gamma } = 0\) otherwise. The *Einstein’s* summation convention over repeated indices is assumed in the sequel. In this way, a contravariant basis can be constructed using the contravariant metric coefficients,

$$\begin{aligned} \mathbf {A}^{\alpha } = A^{\alpha \beta } \mathbf {A}_{\beta } \;, \end{aligned}$$

(10)

where surface normal vector \(\mathbf {A}_3\) stays the same in both the covariant and the contravariant bases. The components of the curvature tensor \(\mathbf {B} = B_{\alpha \beta } \, \mathbf {A}_{\alpha } \otimes \mathbf {A}_{\beta }\) (also known as the *second fundamental form* of a surface) are given by,

$$\begin{aligned} B_{\alpha \beta } = -\mathbf {A}_{3,\alpha } \cdot \mathbf {A}_{\beta } = - \mathbf {A}_{3,\beta } \cdot \mathbf {A}_{\alpha } = \mathbf {A}_3 \cdot \mathbf {A}_{\alpha , \beta } \;, \end{aligned}$$

(11)

which are linked to the curvature along the parametric directions \(\theta _{\alpha }\).

### Mechanics of lightweight structures

Lightweight structures are typically represented by their mid-surface \(\Omega \) which consists of all particles \(\mathbf {X}\) in the reference configuration, see for example in [40]. Such structures comprise membranes and shells which are considered thin, that is, \( \frac{{\bar{h}}}{{\bar{R}}} \ll 20\), \({\bar{h}}\) and \({\bar{R}}\) being the structural thickness and the radius of curvature, respectively. Herein a Lagrangian description of the motion is assumed and the problem is posed on the unknown displacement field \(\mathbf {d} : \Omega \rightarrow \Omega _t\) of the mid-surface, where \(\Omega _t\) stands for the current configuration consisting of all particles \(\mathbf {x} = \mathbf {X} + \mathbf {d}\) at time \(t \in {\mathbb {T}}\) where \({\mathbb {T}} = [0,T_{\infty }]\), \(T_0\) and \(T_{\infty }\) being the start and the end time of the dynamic process. In this way, assumed is that \(\Omega \) and \(\Omega _t\) are represented by a parametric domain \({\hat{\Omega }}\) via the geometric maps (“Differential geometry of surfaces” section) \(\mathbf {S}\) and \(\mathbf {S}_t\), respectively, see Fig. 2. Accordingly, the displacement field may be expressed on both the Cartesian basis \(\mathbf {e}_i\) and a curvilinear basis \(\mathbf {A}_{\alpha }\), \(\mathbf {A}_3\) (see Eq. (8)) as follows,

$$\begin{aligned} \mathbf {d} = d^0_i \mathbf {e}_i = d^{\alpha } \mathbf {A}_{\alpha } + d_3 \mathbf {A}_3 \;. \end{aligned}$$

(12)

The weak form of dynamic equilibrium for these structures can be written as follows: Find \(\mathbf {d} \in \varvec{{\mathcal {H}}}^\alpha (\Omega )\) for each time instance \(t \in {\mathbb {T}}\) such that,

$$\begin{aligned} \left\langle \delta \mathbf {d} , \rho \, {\bar{h}} \, \ddot{\mathbf {d}} \right\rangle _{0,\Omega } + \left\langle \delta \mathbf {d} , c \, {\bar{h}} \, \dot{\mathbf {d}} \right\rangle _{0,\Omega } + a (\delta \mathbf {d} , \mathbf {d}) = l (\delta \mathbf {d}) \;, \quad \forall \delta \mathbf {d} \in \varvec{{\mathcal {H}}}^\alpha (\Omega ) \;, \end{aligned}$$

(13)

where \(\varvec{{\mathcal {H}}}^\alpha (\Omega )\) stands for the space of all square integrable vector-valued functions with square integrable derivatives up to \(\alpha \)-th order in \(\Omega \). Moreover, \(\alpha = 1\) and \(\alpha = 2\) for the membrane and the Kirchhoff-Love shell problem, respectively. This is because the curvature tensor involves second derivatives on the displacement field in Kirchhoff-Love shell analysis, see also in [13]. The first and the second terms in Eq. (13) stand for the inertia and damping of the structure, where \(\rho \) and *c* stand for the structural density and the damping coefficient, respectively. The form *a* is specialized for the membrane and the Kirchhoff-Love shell structural analysis in the following sections. The linear functional \(l:\varvec{{\mathcal {H}}}^\alpha (\Omega ) \rightarrow {\mathbb {R}}\) is defined as,

$$\begin{aligned} l(\delta \mathbf {d}) = \left\langle \delta \mathbf {d} , \mathbf {b} \right\rangle _{0,\Omega } \;, \end{aligned}$$

(14)

where \(\mathbf {b}\) stands for the body forces acting in \(\Omega \). Especially for the membrane and the Kirchhoff-Love shell problems more types of external loads can be considered, see in [17, 41] for more information. The inner product \(\left\langle \bullet , \bullet \right\rangle _{0,\Omega }\) in the \(\mathbf {{\mathcal {L}}}^2 (\Omega )\) space (space of square integrable vector-valued functions in \(\Omega \)) in Eq. (14) is defined as,

$$\begin{aligned} \left\langle \delta \mathbf {d} , \mathbf {b} \right\rangle _{0,\Omega } = \int _{\Omega } \delta \mathbf {d} \cdot \mathbf {b} \, \text {d} \Omega \;. \end{aligned}$$

(15)

Note that the weak form of dynamic equilibrium in Eq. (13) is formulated at each patch \(\Omega ^{(i)}\) independently, not accounting for the Dirichlet boundary conditions along a portion of the domain’s boundary \(\Gamma _{\text {d}} \subset \partial \Omega \). The continuity across the multipatches and the weak application of the Dirichlet boundary conditions are specialized for the membrane and the Kirchhoff-Love shell in the following sections. The weak enforcement of the these constraints is essential as the multiple patches are not conforming along their common interfaces and the Dirichlet boundary conditions are typically enforced along trimming curves where the basis functions are not interpolatory. Thus, the strong enforcement of the interface and boundary constraints is in general inapplicable within IBRA. In the sequel, the dynamic form of weak equilibrium in Eq. (13) is posed on the decomposed open domain \(\Omega _{\text {d}}\) defined in Eq. (7b) and the interface continuity conditions are discussed in the sequel.

### Membrane structural analysis on multipatches

The isogeometric membrane structural analysis on multipatches employed in this work is based on the Penalty and Nitsche-type formulations presented in [41], where also weak application of the Dirichlet boundary conditions is considered. The Nitsche-type formulation is considered as a consistent extension of the Penalty method. This is because the Nitsche-type formulation in its original forms lacks coercivity and Penalty-like stabilization terms are added to restore coercivity. The corresponding stabilization parameters can be estimated by the solution of interface and boundary eigenvalue problems at each time step [16, 41]. On the other hand, one obtains a pure Penalty formulation when leaving only the Penalty-like stabilization terms by excluding the additional Nitsche terms. Therefore the statement of the Nitsche-type formulation includes that of the Penalty formulation and thus both are herein presented in a unified manner. The presented numerical examples of multipatch isogeometric membrane structural analysis using the Penalty method are computed using the IBRA implementation in Carat++ in-house software [42] whereas the ones using the Nitsche-type method are computed using a \(\text {MATLAB}^{\textregistered }\) based framework freely available in [43]. Three-dimensional membranes can not in principle withstand compression without any form of stabilization due to wrinkling which is a type of zero energy mode. Wrinkling enhanced models have been extensively studied in the literature, see also in [44]. Additionally, membranes typically need to be under prestress in order to avoid wrinkling and be rendered stable. The latter results in a non-trivial design in that not every free-form shape may render a shape of static equilibrium. For this purpose *form-finding* methods have been developed [45] and in particular the *Updated Reference Strategy*, (URS) see in [46, 47]. In this study, membranes in their original design are considered and moreover no form-finding is used for the herein presented numerical examples as the chosen geometries are by construction compatible with the applied prestress while no cables are embedded, see also in [21, 48] for more information. The *Green-Lagrange* (GL) strain tensor of the mid-surface \(\varvec{\varepsilon } = \varepsilon _{\alpha \beta } \, \mathbf {A}^{\alpha } \otimes \mathbf {A}^{\beta }\) is employed and its components are given by,

$$\begin{aligned} \varepsilon _{\alpha \beta } = \frac{1}{2} \left( a_{\alpha \beta } - A_{\alpha \beta } \right) \;, \end{aligned}$$

(16)

\(a_{\alpha \beta } = \mathbf {a}_{\alpha }\cdot \mathbf {a}_{\beta }\) being the covariant metric coefficients and \(\mathbf {a}_{\alpha }\) the base vectors of the current configuration. Moreover, \(\mathbf {A}^{\alpha }\) stand for the contravariant base vectors of the reference configuration. The components of the energetically conjugate *2nd Piola-Kirchhoff* (PK2) stress-resultant force tensor \(\mathbf {n} = n^{\alpha \beta } \, \mathbf {A}_{\alpha } \otimes \mathbf {A}_{\beta }\) of the mid-surface are defined by means of the linear *Hooke’s* law (*Saint-Venant* material), namely,

$$\begin{aligned} n^{\alpha \beta } = {\mathcal {C}}^{\alpha \beta \gamma \delta } \varepsilon _{\gamma \delta } \;, \end{aligned}$$

(17)

where the components of the material tensor \(\varvec{{\mathcal {C}}} = {\mathcal {C}}^{\alpha \beta \gamma \delta } A_{\alpha } \otimes \mathbf {A}_{\beta } \otimes \mathbf {A}_{\gamma } \otimes \mathbf {A}_{\delta }\) are given by,

$$\begin{aligned} {\mathcal {C}}^{\alpha \beta \gamma \delta } = \frac{E\,{\bar{h}}}{2 \left( 1 + \nu \right) } \left( A^{\alpha \gamma } A^{\beta \delta } + A^{\alpha \delta } A^{\beta \gamma } + \frac{2 \nu }{1 - 2 \nu } A^{\alpha \beta } A^{\gamma \delta } \right) \;, \end{aligned}$$

(18)

*E* and \(\nu \) being the *Young’s* (elastic) modulus and the *Poisson’s* ratio, respectively. The traction along any curve \(\gamma \) on surface \(\Omega \) is defined by [38],

$$\begin{aligned} \mathbf {t} = (n^{\alpha \beta } + n_0^{\alpha \beta }) u_{\alpha } \mathbf {a}_{\beta } \;, \end{aligned}$$

(19)

where \(u_{\alpha }\) stand for the covariant components of the curve’s \(\gamma \) normal vector \(\mathbf {u}\) on surface \(\Omega \) and where \(n_0^{\alpha \beta }\) stand for the contravariant coefficients of the prestress tensor \(\mathbf {n}_0\). Concerning the multipatch formulation, the solution space for the Nitsche-type and the Penalty methods is \(\varvec{{\mathcal {V}}} = \varvec{{\mathcal {H}}}^1 (\Omega _{\text {d}}) \cup \varvec{{\mathcal {H}}}^1 (\gamma _{\text {i}}) \cup \varvec{{\mathcal {H}}}^1 (\Gamma _{\text {d}})\) and \(\varvec{{\mathcal {V}}} = \varvec{{\mathcal {H}}}^1 (\Omega _{\text {d}}) \cup \varvec{{\mathcal {L}}}^2 (\gamma _{\text {i}}) \cup \varvec{{\mathcal {L}}}^2 (\Gamma _{\text {d}})\), respectively. Fields restricted in a patch and along an interface are represented in the sequel by a superscript that is, \(\bullet _{|_{\Omega ^{(i)}}} = \bullet ^{(i)}\) and \(\bullet _{|_{\gamma ^{(i,j)}}} = \bullet ^{(i,j)}\), respectively. Let \(\hat{\varvec{\chi }}\) stand for the interface displacement jump, that is, \(\hat{\varvec{\chi }}_{|_{\gamma _{\text {i}}^{(i,j)}}} = \mathbf {d}^{(i)} - \mathbf {d}^{(j)}\). The mean interface traction field is given by,

$$\begin{aligned} \bar{\mathbf {t}} = \frac{1}{2} \left( \mathbf {t}^{(i)} - \mathbf {t}^{(j)} \right) \;. \end{aligned}$$

(20)

Accordingly, form \(a: \varvec{{\mathcal {V}}} \times \varvec{{\mathcal {V}}} \rightarrow {\mathbb {R}}\) in Eq. (13) is defined as follows for the multipatch isogeometric membrane analysis using the Nitsche-type method,

$$\begin{aligned} \begin{aligned} a(\delta \mathbf {d} , \mathbf {d}) = \int _{\Omega _{\text {d}}} \delta \varvec{\varepsilon } : ( \mathbf {n} + \mathbf {n}_0 ) \, \text {d} \Omega - \left\langle \delta \hat{\varvec{\chi }} , \bar{\mathbf {t}} \right\rangle _{0,\gamma _{\text {i}}} - \left\langle \delta \bar{\mathbf {t}} , \hat{\varvec{\chi }} \right\rangle _{0,\gamma _{\text {i}}} \\ + \left\langle \delta \hat{\varvec{\chi }} , {\hat{\alpha }} \hat{\varvec{\chi }} \right\rangle _{0,\gamma _{\text {i}}} - \left\langle \delta \mathbf {d} , \mathbf {t} \right\rangle _{\Gamma _{\text {d}}} - \left\langle \delta \mathbf {t} , \mathbf {d} \right\rangle _{\Gamma _{\text {d}}} + \left\langle \delta \mathbf {d} , {\bar{\alpha }} \mathbf {d} \right\rangle _{0,\Gamma _{\text {d}}} \;, \end{aligned} \end{aligned}$$

(21)

where \({\hat{\alpha }}:\gamma _{\text {i}} \rightarrow {\mathbb {R}}\) and \({\bar{\alpha }}:\Gamma _{\text {d}} \rightarrow {\mathbb {R}}\) stand for the stabilization parameters (Nitsche-type method) or the Penalty parameters (Penalty method) when the additional terms stemming from the Nitsche-type method are omitted. In case the Nitsche-type method is employed, the corresponding stabilization parameters are estimated automatically by solving a sequence of interface and boundary eigenvalue problems, see in [41] for more information. There are defined as piecewise constant along each interface \(\gamma _{\text {i}}^{(i,j)}\) and each Dirichlet boundary \(\Gamma _{\text {d}}^{(i)}\). On the other hand, in case the Penalty method is employed the Penalty parameters are discretization-dependent and are computed similar to the rule proposed in [17], namely,

$$\begin{aligned} {\hat{\alpha }}^{(i,j)}&= h_{-1}^{(i,j)} \left\| \varvec{{\mathcal {C}}} \right\| \;, \end{aligned}$$

(22a)

$$\begin{aligned} {\bar{\alpha }}^{(i)}&= h_{-1}^{(i)} \left\| \varvec{{\mathcal {C}}} \right\| \;, \end{aligned}$$

(22b)

where \(h_{-1}^{(i,j)}\) and \(h_{-1}^{(i)}\) stand for the inverse of the smallest knot span length in the physical space along the interface trimming curve \(\gamma _{\text {i}}^{(i,j)}\) and along the trimming curve \(\Gamma _{\text {d}}^{(i)}\) defining the portion of the Dirichlet boundary having an intersection with \(\partial \Omega ^{(i)}\) within an isogeometric discretization, respectively. The norm of the material tensor in Eqs. (22) is understood as \(\left\| \varvec{{\mathcal {C}}} \right\| = \left( \sum _{\alpha = 1}^2 \sum _{\beta = 1}^2 \sum _{\gamma = 1}^2 \sum _{\delta = 1}^2 \left( {\mathcal {C}}^{\alpha \beta \gamma \delta } \right) ^2 \right) ^{1/2}\), that is, the square root of the sum of its squared components.

### Kirchhoff-Love structural analysis on multipatches

Similar to “Membrane structural analysis on multipatches” section, the herein employed isogeometric Kirchhoff-Love shell structural analysis on multipatches accounting for weak Dirichlet boundary conditions is based on a Penalty formulation as presented in [17, 20]. The employed numerical example of multipatch isogeometric Kirchhoff-Love shell structural analysis using the Penalty method, that of the NREL phase VI wind turbine [23], is computed using the IBRA implementation within the in-house software Carat++. Moreover, small strains are herein assumed and thus the corresponding linearised theory is briefly presented.

The linearised GL strain strain tensors \(\varvec{\varepsilon } = \varepsilon _{\alpha \beta } \, \mathbf {A}^{\alpha } \otimes \mathbf {A}^{\beta }\) and \(\varvec{\kappa } = \kappa _{\alpha \beta } \mathbf {A}^{\alpha } \otimes \mathbf {A}^{\beta }\) for the membrane and the bending strain are defined as [49],

$$\begin{aligned} \varepsilon _{\alpha \beta }&= \frac{1}{2} \left( \mathbf {A}_{\beta } \cdot \mathbf {d}_{,\alpha } + \mathbf {A}_{\alpha } \cdot \mathbf {d}_{,\beta } \right) \;, \end{aligned}$$

(23a)

$$\begin{aligned} \kappa _{\alpha \beta }&= - \mathbf {A}_3 \cdot \mathbf {d}_{,\alpha \beta } + \mathbf {A}_{\alpha ,\beta } \cdot \mathbf {A}_3 \frac{1}{{\bar{j}}} \left( \left( \mathbf {A}_2 \times \mathbf {A}_3 \right) \cdot \mathbf {d}_{,1} - \left( \mathbf {A}_1 \times \mathbf {A}_3 \right) \cdot \mathbf {d}_{,2} \right) \nonumber \\&+\frac{1}{{\bar{j}}} \left( \left( \mathbf {A}_{\alpha ,\beta } \times \mathbf {A}_2 \right) \cdot \mathbf {d}_{,1} - \left( \mathbf {A}_{\alpha ,\beta } \times \mathbf {A}_1 \right) \cdot \mathbf {d}_{,2} \right) \;, \end{aligned}$$

(23b)

where \((\bullet )_{,\alpha \beta } = \partial (\bullet )_{,\alpha }/\partial \theta _{\beta }\). The PK2 stress-resultant force tensor for the in-plane stiffness of the Kirchhoff-Love shell is defined as in Eq. (17). Similarly, the PK2 stress-resultant tensor for bending stiffness of the Kirchhoff-Love shell \(\mathbf {m} = m^{\alpha \beta } \, \mathbf {A}_{\alpha } \otimes \mathbf {A}_{\beta }\) is defined using also the linear Hooke’s law, that is,

$$\begin{aligned} m^{\alpha \beta } = {\bar{h}}^2\,{\mathcal {C}}^{\alpha \beta \gamma \delta } \kappa _{\gamma \delta } \;. \end{aligned}$$

(24)

The rotation field \(\varvec{\omega } = \omega ^{\zeta } \mathbf {A}_{\zeta }\) needs to be in this case defined, namely,

$$\begin{aligned} \omega ^{\zeta } = - \left( d_{3,\alpha } + d^{\gamma } B_{\gamma \alpha } \right) \epsilon ^{\alpha \zeta } \;, \end{aligned}$$

(25)

where \(\epsilon ^{\alpha \zeta }\) is the *Levi-Civita* symbol. For the multipatch formulation using the Penalty method, the solution space is in this case \(\varvec{{\mathcal {V}}} = \varvec{{\mathcal {H}}}^2 (\Omega _{\text {d}}) \cup \varvec{{\mathcal {H}}}^1 (\gamma _{\text {i}}) \cup \varvec{{\mathcal {L}}}^2 (\Gamma _{\text {d}})\). Let \(\tilde{\varvec{\chi }}\) stand for the jump on the rotation field across the multipatches, that is, \(\tilde{\varvec{\chi }}_{|_{\gamma _{\text {i}}^{(i,j)}}} = \varvec{\omega }^{(i)} + \varvec{\omega }^{(j)}\). In this way, the form \(a:\varvec{{\mathcal {V}}} \times \varvec{{\mathcal {V}}} \rightarrow {\mathbb {R}}\) in Eq. (13) for the multipatch isogeometric Kirchhoff-Love shell analysis using the Penalty method is defined as,

$$\begin{aligned} \begin{aligned} a (\delta \mathbf {d} , \mathbf {d}) = \int _{\Omega _{\text {d}}} \delta \varvec{\varepsilon } : \mathbf {n} + \delta \varvec{\kappa } : \mathbf {m} \, \text {d} \Omega + \left\langle \delta \hat{\varvec{\chi }} , {\hat{\alpha }} \hat{\varvec{\chi }} \right\rangle _{0,\gamma _{\text {i}}} + \left\langle \delta \tilde{\varvec{\chi }} , {\tilde{\alpha }} \tilde{\varvec{\chi }} \right\rangle _{0,\gamma _{\text {i}}} \\ +\left\langle \delta \mathbf {d} , {\bar{\alpha }} \mathbf {d} \right\rangle _{0,\Gamma _{\text {d}}} \;. \end{aligned} \end{aligned}$$

(26)

Additionally, \({\tilde{\alpha }}:\gamma _{\text {i}} \rightarrow {\mathbb {R}}\) stands for the Penalty parameter associated with the imposition of the rotation continuity across the interfaces. It is chosen also piecewise constant and is defined similar to Eqs. 22, that is,

$$\begin{aligned} {\tilde{\alpha }}^{(i,j)} = h_{-1}^{(i,j)} \, h^2 \Vert \varvec{{\mathcal {C}}} \Vert \;, \end{aligned}$$

(27)

along interface boundary \(\gamma _{\text {i}}^{(i,j)}\).

### Isogeometric spatial discretization on trimmed multipatches

Concerning the discretization of the aforementioned weak forms, the *Isogeometric B-Rep Analysis* (IBRA) is employed, see also in [20]. In this way, the finite dimensional subspace \(\varvec{{\mathcal {V}}}_{\text {h}} \subset \varvec{{\mathcal {V}}}\) is constructed using the parametric description of each patch \(\Omega ^{(i)}\) as \(\varvec{{\mathcal {V}}}_{\text {h}} = \prod _{i = 1}^{n_{\text {s}}} \varvec{{\mathcal {V}}}_{\text {h}}^{(i)}\) where,

$$\begin{aligned} \varvec{{\mathcal {V}}}_{\text {h}}^{(i)} = \left\{ \mathbf {d}^{(i)} \in \varvec{{\mathcal {V}}}^{(i)} \left| \mathbf {d}^{(i)} \in \varvec{{\mathcal {R}}} ( \Omega ^{(i)}) \; \forall i = 1,\ldots ,n_{\text {s}} \right. \right\} \;, \end{aligned}$$

(28)

\(\varvec{{\mathcal {R}}} ( \Omega ^{(i)} )\) being the space of all vector-valued piecewise rational polynomials for which the NURBS basis functions of the geometric parametrization constitute a basis in each patch \(\Omega ^{(i)}\). Let \(\bar{\varvec{\phi }}^{(i)}_j\), with \(j = 1,\ldots , \dim \varvec{{\mathcal {V}}}^{(i)}_{\text {h}}\), be a basis of \(\varvec{{\mathcal {V}}}^{(i)}_{\text {h}}\) for all \(i = 1,\ldots ,n_{\text {s}}\). Then, there exist reals \({\hat{d}}^{(i)}_j\), the so called *Degrees of Freedom* (DOFs), such that for each \(\mathbf {d} \in \varvec{{\mathcal {V}}}_{\text {h}}\) it holds,

$$\begin{aligned} \mathbf {d} = \sum _{i = 1}^{n_{\text {s}}} \sum _{j = 1}^{\dim \varvec{{\mathcal {V}}}^{(i)}_{\text {h}}} \bar{\varvec{\phi }}^{(i)}_j {\hat{d}}^{(i)}_j \;. \end{aligned}$$

(29)

Herein, the vector-valued NURBS basis functions are constructed as,

$$\begin{aligned} \bar{\varvec{\phi }}^{(i)}_r = R_{{\hat{p}}_1^{(i)},{\hat{p}}_2^{(i)},k}^{(i)} \, \mathbf {e}_l \;, \end{aligned}$$

(30)

where \(k = \lceil \frac{r}{3} \rceil \) and \(l = r -3 \lceil \frac{r}{3} \rceil + 3\) for all \(r = 1,\ldots \dim \varvec{{\mathcal {V}}}^{(i)}_{\text {h}}\) stand for the indices of the control points and the Cartesian directions, respectively. Additionally, \(R_{{\hat{p}}_1^{(i)},{\hat{p}}_2^{(i)},k}^{(i)}\) and \(n_\alpha ^{(i)}\), stand for the scalar-valued NURBS basis functions in patch \(\Omega ^{(i)}\) with polynomial orders \({\hat{p}}_1^{(i)}\) and \({\hat{p}}_2^{(i)}\) and the number of control points of patch \(\Omega ^{(i)}\) in \(\theta ^{(i)}_\alpha \)-parametric direction, respectively, see “Non-uniform rational b-spline surfaces” section. The latter implies that \(\dim \varvec{{\mathcal {V}}}^{(i)}_{\text {h}} = 3 n_1^{(i)}n_2^{(i)}\). These DOFs do not represent physical values since they are defined on the control points which in general do not interpolate the geometry.

In this way, projection of variational problem in Eq. (13) onto \(\varvec{{\mathcal {V}}}_{\text {h}}\) results into the following discretized in space equation system,

$$\begin{aligned} \mathbf {M} \ddot{\hat{\mathbf {d}}} + \mathbf {D} \dot{\hat{\mathbf {d}}} + \mathbf {R} ( \hat{\mathbf {d}} ) = \mathbf {F}_t \;, \end{aligned}$$

(31)

where \(\ddot{\hat{\mathbf {d}}}\), \(\dot{\hat{\mathbf {d}}}\) and \(\hat{\mathbf {d}}\) stand for the vectors of acceleration, velocity and displacement DOFs, respectively. In addition, \(\mathbf {M}\) and \(\mathbf {D}\) stand for the mass and damping matrices resulting from the spatial discretization of the first and second terms of variational problem in Eq. (13), respectively. Moreover, \(\mathbf {R} ( \hat{\mathbf {d}} )\) stands for the steady-state residual vector whose linearization results in the steady-state tangent stiffness matrix \(\mathbf {K}( \hat{\mathbf {d}} )\) and whose entries are given by \(K_{ij} ( \hat{\mathbf {d}} ) = \frac{\partial R_i ( \hat{\mathbf {d}} )}{\partial {\hat{d}}_j} \), \(R_i (\hat{\mathbf {d}})\) and \({\hat{d}}_j\) being the *i*-th component of the residual vector and the *j*-th DOF, respectively. The definition of the tangent stiffness matrices for the membrane BVP can be found in [21, 41] and for the Kirchhoff-Love shell BVP in [17, 20]. In this study, the damping matrix is approximated using the Rayleigh damping method, that is,

$$\begin{aligned} \mathbf {D} = \alpha _{\text {r}} \mathbf {M} + \beta _{\text {r}} \mathbf {K}(\hat{\mathbf {d}}_0) \;, \end{aligned}$$

(32)

where \(\alpha _{\text {r}}\) and \(\beta _{\text {r}}\) stand for the so-called *Rayleigh* damping parameters and \(\hat{\mathbf {d}}_0\) stands for the initial condition on the displacement field, see in [50] for more information.

### Time discretization and modal analysis

The Newmark method [51] is used in this study for the time discretization of linear equation system in Eq. (31). Accordingly, the continuous time domain \({\mathbb {T}}\) is discretized into a set of time steps \(t_{\hat{n}}\). The system is linearised using the Newton-Raphson iterative method, that is,

$$\begin{aligned} \bar{\mathbf {K}}_{{\hat{n}},{\hat{i}}} \; \Delta _{{\hat{i}}} \hat{\mathbf {d}}_{{\hat{n}}} = - \bar{\mathbf {R}}_{{\hat{n}},{\hat{i}}} \;, \end{aligned}$$

(33)

where \(\Delta _{{\hat{i}}} \, \hat{\mathbf {d}}_{{\hat{n}}} = \hat{\mathbf {d}}_{{\hat{n}},{\hat{i}} + 1} - \hat{\mathbf {d}}_{{\hat{n}},{\hat{i}}}\), \(\hat{\mathbf {d}}_{{\hat{n}},{\hat{i}}}\) being the vector of DOFs at the \({\hat{n}}\)-th time step and at \({\hat{i}}\)-th Newton-Raphson iteration. The dynamic stiffness matrix \(\bar{\mathbf {K}}_{{\hat{n}},{\hat{i}}}\) and residual vector \(\bar{\mathbf {R}}_{{\hat{n}},{\hat{i}}}\) at the \({\hat{n}}\)-th time step and at \({\hat{i}}\)-th Newton-Raphson iteration are defined by means of the corresponding steady state tangent stiffness matrix \(\mathbf {K}_{{\hat{n}},{\hat{i}}}\) and residual vector \(\mathbf {R} _{{\hat{n}},{\hat{i}}}\), namely [35, 41, 52],

$$\begin{aligned} \bar{\mathbf {K}}_{{\hat{n}},{\hat{i}}}&= \left( \frac{1}{\beta _{\text {n}} \left( \Delta t \right) ^2} \mathbf {M} + \frac{\gamma _{\text {n}}}{\beta _{\text {n}} \Delta t} \mathbf {D} \right) + \mathbf {K}_{{\hat{n}},{\hat{i}}} \;, \end{aligned}$$

(34a)

$$\begin{aligned} \bar{\mathbf {R}}_{{\hat{n}},{\hat{i}}}&= \left( \frac{1}{\beta _{\text {n}} \left( \Delta t \right) ^2} \mathbf {M} + \frac{\gamma _{\text {n}}}{\beta _{\text {n}} \Delta t} \mathbf {D}\right) \hat{\mathbf {d}}_{{\hat{n}},{\hat{i}}} + \mathbf {R}_{{\hat{n}},{\hat{i}}} - \hat{\mathbf {F}}_{{\hat{n}}} \nonumber \\&-\left( \frac{1}{\beta _{\text {n}} \left( \Delta t \right) ^2} \mathbf {M} + \frac{\gamma _{\text {n}}}{\beta _{\text {n}} \Delta t}\mathbf {D} \right) \hat{\mathbf {d}}_{{\hat{n}} - 1} -\left( \frac{1}{\beta _{\text {n}} \Delta t} \mathbf {M} - \frac{\beta _{\text {n}} - \gamma _{\text {n}}}{\beta _{\text {n}}} \right) \dot{\hat{\mathbf {d}}}_{{\hat{n}} - 1} \nonumber \\&-\left( \frac{1 - 2 \beta _{\text {n}}}{2 \beta _{\text {n}}} \mathbf {M} - \Delta t \frac{2 \beta _{\text {n}} - \gamma _{\text {n}}}{2 \beta _{\text {n}}} \mathbf {D} \right) \ddot{\hat{\mathbf {d}}}_{{\hat{n}} - 1} \;, \end{aligned}$$

(34b)

where \(\beta _{\text {n}}\) and \(\gamma _{\text {n}}\) stand for the Newmark parameters and where \(\hat{\mathbf {d}}_{{\hat{n}} - 1,{\hat{i}}}\), \(\dot{\hat{\mathbf {d}}}_{{\hat{n}} - 1,{\hat{i}}}\) and \(\ddot{\hat{\mathbf {d}}}_{{\hat{n}} - 1,{\hat{i}}}\) stand for the displacement, velocity and acceleration DOFs at time step \(t_{{\hat{n}} - 1}\), see in [41] for more information on the discrete equation systems.

Concerning modal analysis, this is performed on the linearised system using the linear stiffness matrix \(\mathbf {K}(\hat{\mathbf {d}}_0)\) and by solving the following eigenvalue problem,

$$\begin{aligned} \det \left( \omega _i^2 \mathbf {M} + \mathbf {K}(\hat{\mathbf {d}}_0) \right) = 0 \;, \end{aligned}$$

(35)

where \(\omega _i = 2 \pi f_i\) are the circular eigenfrequencies and where \(f_i\) stand for the natural eigenfrequencies of the system.

### Isogeometric B-Rep analysis of the NREL phase VI wind turbine

In this section, the NREL phase VI wind turbine with flexible blades [23] is employed as demonstration of isogeometric analysis on multipatch surfaces in industrial scale applications, see Fig. 3c. This numerical example is herein employed for the demonstration of IBRA on a real-world engineering structure in multiphysics environment and for validating the underlying computational models which are later on used in the context of FSI with the proposed isogeometric B-Rep mortar-based mapping method. A picture of the actual turbine can be seen in Fig. 3a. The corresponding CAD model consisting of rigid parts and the two flexible blades whose stiffness is enhanced using two longitudinal spars along the longitudinal trimming curves on the blades’ surfaces, is shown in Fig. 3b. The problem is solved using the linearised Kirchhoff-Love shell theory within IBRA presented in “Isogeometric lightweight structural analysis on trimmed multipatches” section. The results of this simulation were firstly presented in the dissertation [35] and are repeated herein for the sake of completeness of this study.

The original computational model in [53] involves a composite material model with varying thickness by analysing the data provided in [23]. Herein a simplified model is used with a Saint-Venant Kirchhoff material. The homogenized Young’s modulus, density and thickness of the flexible blades are obtained by a calibration using a geometrically linear static and a modal analysis against the maximum displacement and the first eigenfrequency, respectively, computed in [53]. In this way, the Young’s modulus, the density and the thickness of the flexible blades assumed to be \(E = 6 \times 10^{10}\) Pa, \(\rho = 1.515 \times 10^3\; \text {Kg/m}^3\) and \({\bar{h}} = 7\) mm, respectively. The Poisson ratio is then chosen as \(\nu = 0.2\). Regarding the static analysis, the flexible blades are subject to self weight, namely, \(\mathbf {b} = - \rho {\bar{h}} \, \mathbf {e}_3\). The results of IBRA for this example are compared with the results obtained using a standard finite element discretization of the flexible blades, see Fig. 3. Accordingy, the FEM model consists of 48630 triangular elements (Fig. 3(c)) based on a shell model with *Reissner-Mindlin* (RM) kinematics within Carat++ software ( [42]). Then, the corresponding h-refined multipatch NURBS computational model of the flexible blades is shown in Fig. 3(d). Subsequently, the NURBS computational model of the right blade is shown both intact and decomposed into its underlying trimmed patches in Fig. 4 where the geometric complexity and the large number of the underlying trimmed NURBS multipatches comprising the geometry is highlighted. It is worth mentioning that the spars and the tip of the NURBS computational model are connected to the rest of the blades’ skin with a \(C^0\)-parametric continuity forming geometric kinks, thus adding another complexity to the NURBS multipatch model. Each blade consists of 37 trimmed patches with 170 interface boundaries of highly diverse sizes and parametrizations. The scaling associated to the Penalty parameters is then chosen as the inverse of the minimum element edge size along each interface and Dirichlet boundary (see “Kirchhoff-Love structural analysis on multipatches” section) for \({\hat{\alpha }},{\tilde{\alpha }}\) and \({\bar{\alpha }}\), respectively.

The contour of the 2-norm of the displacement field \(\Vert \mathbf {d} \Vert _2\) across the blades in the current configuration due to self-weight for both the standard finite element analysis and IBRA is shown in Fig. 5 demonstrating excellent accordance of the results. Moreover, an eigenfrequency analysis for both models is performed, see Eq. (35), and the first three eigenfrequencies of both the standard FEM and IGA models are shown in Fig. 6 demonstrating once more an excellent accordance of the results also in this context.