The geometric approximation inherent in the mesh of the traditional FEA can lead to accuracy problems [18], especially for surfaces. As a result, the value of \(\varepsilon_{{\text{rms}}}^{F}\) is notoriously sensitive to geometric discretization. Contrarily, the IGA is geometrically exact no matter how coarse the discretization is by using the functions from the geometry description as basis functions for the analysis. The displacement field of the IGA element *e* can be described as

$$\tilde{\varvec{u}}_{e} \left(\varvec{\xi}\right) = \sum\limits_{a = 1}^{n} {R_{e}^{a} \left(\varvec{\xi}\right)} \varvec{u}_{e}^{a} = \left( {\varvec{u}_{e} } \right)^{\text{T}} \varvec{R}_{e} ,$$

(6)

where \(R_{e}^{a} \left(\varvec{\xi}\right)\) denotes the basis function of the *a*th control point of element *e*, and \(\varvec{u}_{e}^{a}\) denotes the displacement of the control point. Fig. 2. shows the IGA element as knot spans, where the red lines denote the knot spans and the black dots the control points of element *e*.

Unlike the Lagrange elements, the IGA elements are taken to be knot spans, namely, [*ξ*
_{
i−1}, *ξ*
_{
i
}] × [*η*
_{
i−1}, *η*
_{
i
}], and the control points are not always located in the element.

### 3.1 Element formulation for NURBS-based IGA

Shell and beam elements are required for the surface error analysis of the antenna reflector. A brief introduction of the rotation-free Kirchhoff-Love shell element based on NURBS is presented in this section. In addition, a Euler-Bernoulli beam element of three degrees of freedom based on Bézier extraction, which maps the Bernstein polynomial basis on Bézier elements to the NURBS basis, is developed.

#### 3.1.1 Kirchhoff-Love Shell element

Kirchhoff-Love shell based on NURBS has been presented by KIENDL, et al [28]. The variation of the internal work formula of Kirchhoff-Love theory

$$\begin{aligned} \delta W_{\text{int}} & = \int_{\varOmega } {\left( {t\varepsilon_{\gamma \delta } C^{\alpha \beta \gamma \delta } \delta \varepsilon_{\alpha \beta } + \frac{{t^{3} }}{12}\kappa_{\gamma \delta } C^{\alpha \beta \gamma \delta } \delta \kappa_{\alpha \beta } } \right)} \text{d}\varOmega , \\ \varepsilon_{\alpha \beta } & = \frac{1}{2}\left( {\bar{\varvec{a}}_{\alpha } \cdot \varvec{u}_{,\beta } + \bar{\varvec{a}}_{\beta } \cdot \varvec{u}_{,\alpha } + \varvec{u}_{,\alpha } \cdot \varvec{u}_{,\beta } } \right), \\ \kappa_{\alpha \beta } & = \bar{\varvec{a}}_{\alpha ,\beta } \cdot \bar{\varvec{a}}_{3} - \varvec{a}_{\alpha ,\beta } \cdot \varvec{a}_{3} = \bar{\varvec{a}}_{\alpha ,\beta } \cdot \bar{\varvec{a}}_{3} - \bar{\varvec{a}}_{\alpha ,\beta } \cdot \varvec{a}_{3} - \varvec{u}_{,\alpha \beta } \cdot \varvec{a}_{3} , \\ \end{aligned}$$

(7)

where *C*
^{αβγδ} denotes the elasticity tensor, *ε*
_{
αβ
} denotes the membrane strain, *κ*
_{
αβ
} denotes the bending strain, \(\bar{\varvec{a}}_{\alpha }\) denotes the basis vector of middle surface in the reference configuration, *u* is the displacement of middle surface, and the subscript ‘,*α*’ denotes the derivative with respect to *ξ*
^{α}, *α*=1, 2.

The stiffness matrix of the thin-shell element can be written as

$$\begin{aligned} \varvec{K} & = \sum\limits_{e} {\varvec{K}_{e} = } \sum\limits_{e} {\left[ {\int_{A} {\left( {\frac{Et}{{1 - v^{2} }}\left( {\varvec{D}_{e}^{m} } \right)^{\text{T}} \varvec{FD}_{e}^{m} } \right.} } \right.} \\ & \left. {\left. {\quad + \frac{{Et^{3} }}{{12\left( {1 - v^{2} } \right)}}\left( {\varvec{D}_{e}^{b} } \right)^{\text{T}} \varvec{FD}_{e}^{b} } \right)\text{d}A} \right], \\ \end{aligned}$$

(8)

where *F* is the transformation matrix which links the reference configuration to the deformed configuration, \(\varvec{D}_{e}^{m}\) and \(\varvec{D}_{e}^{b}\) denote the matrix for membrane and bending strains respectively, for details we refer the reader to BEER, et al [36].

#### 3.1.2 Euler-Bernoulli Beam element

For a 3D beam suffered several different loads, there is additionally the assumption that the beam behaves elastically for the combined loads, as well as for the individual loads, and the deflection is small. In this case, the deflection at any point on the beam is simply the sum of the deflections caused by each of the individual loads. We developed an IGA beam loaded in such a manner that the resultant force passes through the longitudinal shear center axis, i.e. no torsion will occur.

As shown in Fig. 3, each node has five parameters {*u*, *v*,

*w*, *θ*
_{
y
}, *θ*
_{
z
}}, where the slope *θ* can be eliminated by adopting standard structural-mechanics notations

$$\left\{ {\begin{array}{*{20}l} {\theta_{y} = - \frac{{\text{d}{\it {w}}}}{{\text{d}{\it {x}}}} = - w^{\prime},} \hfill \\ {\theta_{z} = - \frac{{\text{d}v}}{{\text{d}x}} = - v^{\prime},} \hfill \\ \end{array} } \right.{\kern 1pt}$$

(9)

where the prime symbol (•)′ indicates a derivative with respect to *x*. The geometric equations of strains can be written as

$$\left\{ {\begin{array}{*{20}l} {\varepsilon_{1} \left( x \right) = u\left( x \right)^{\prime } ,} \hfill \\ \begin{aligned} \varepsilon_{2} \left( x \right) = - \hat{y}v\left( x \right)^{\prime \prime } , \hfill \\ \varepsilon_{3} \left( x \right) = - \hat{z}w\left( x \right)^{\prime \prime } , \hfill \\ \end{aligned} \hfill \\ \end{array} } \right.{\kern 1pt}$$

(10)

where *ε*
_{1} is the tensile strain and the others bending strains. The variation of the internal work formula of the beam can be obtained by using the superposition method

$$\begin{aligned} \delta W_{\text{int}} & = \sum\limits_{a = 1}^{3} {\int {E\updelta\varepsilon_{a} \left( x \right)\varepsilon_{a} \left( x \right)} } \\ & {\kern 1pt} = E\left( {A\int_{0}^{l} {\delta u\left( x \right)^{\prime } u\left( x \right)^{\prime } \text{d}x + } } \right.I_{z} \int_{0}^{l} {\delta v\left( x \right)^{\prime \prime } v\left( x \right)^{\prime \prime } \text{d}x} \\ & \quad + {\kern 1pt} \left. {I_{y} \int_{0}^{l} {\delta w\left( x \right)^{\prime \prime } w\left( x \right)^{\prime \prime } \text{d}x} } \right), \\ \end{aligned}$$

(11)

where *E* denotes the Young’s modulus, *A* the cross-sectional area, *I*
_{
y
} and *I*
_{
z
} the second moment of inertia. Substituting Eq. 6 into the internal work formula, we can obtain the stiffness matrix *K* in the local coordinates

$$\begin{aligned} K & = \sum\limits_{e} {K_{e} } \\ & = E\sum\limits_{e} {\left( {\begin{array}{*{20}c} {A\int_{0}^{l} {\left( {\varvec{R}_{e}^{{\prime }} } \right)^{\text{T}} \varvec{R}_{e}^{'} \text{d}x} } & {} & {} \\ {} & {I_{z} \int_{0}^{l} {\left( {\varvec{R}_{e}^{{\prime \prime }} } \right)^{\text{T}} \varvec{R}_{e}^{{\prime \prime }} \text{d}x} } & {} \\ {} & {} & {I_{y} \int_{0}^{l} {\left( {\varvec{R}_{e}^{{\prime \prime }} } \right)^{\text{T}} \varvec{R}_{e}^{{\prime \prime }} \text{d}x} } \\ \end{array} } \right).} \\ \end{aligned}$$

(12)

The NURBS domain can be rewritten in terms of the Bernstein basis by extracting the linear operator which maps the Bernstein polynomial basis on Bézier elements to the NURBS basis

$$\varvec{R}_{e} \left(\varvec{\xi}\right) = \frac{{\varvec{w}^{e} \varvec{C}^{e} \varvec{B}\left(\varvec{\xi}\right)}}{{\left[ {\left( {\varvec{C}^{e} } \right)^{\text{T}} \bar{\varvec{w}}} \right]^{\text{T}} \varvec{B}\left(\varvec{\xi}\right)}},$$

(13)

where *C*
^{e} denotes the Bézier extraction operator of element *e* [37], *ξ* = (*ξ*, *η*) the parametric coordinates defined over the interval [−1, 1], *B*(*ξ*) the Bernstein polynomial basis, \(\bar{\varvec{w}}\) and *w*
^{e} are two expressions for the weights of control points

$$\bar{\varvec{w}} = \left( {\begin{array}{*{20}c} {w_{1} } \\ {w_{2} } \\ \vdots \\ {w_{n} } \\ \end{array} } \right),\;\;\varvec{w}^{e} = \left( {\begin{array}{*{20}c} {w_{1} } & 0 & 0 & 0 \\ 0 & {w_{2} } & 0 & 0 \\ 0 & 0 & \ddots & 0 \\ 0 & 0 & 0 & {w_{n} } \\ \end{array} } \right).$$

(14)

Additionally, transformation of coordinates to a common global system, which will be denoted by \(\bar{x}\bar{y}\bar{z}\) with the local system *xyz*, will be necessary to assemble the elements. For an element contains *n* control points, a transformation matrix *T*
^{e} is given to transform the forces and displacements from the global to the local system

$$\mathop {\varvec{T}^{e} }\limits_{{\left( {3n \times 3n} \right)}} = \left( {\begin{array}{*{20}c}\varvec{\lambda}& {} & {} & 0 \\ {} &\varvec{\lambda}& {} & {} \\ {} & {} & \ddots & {} \\ 0 & {} & {} &\varvec{\lambda}\\ \end{array} } \right),$$

(15)

with *λ* being a 3×3 matrix of direction cosines between the two sets of axes

$$\varvec{\lambda}= \left( {\begin{array}{*{20}c} {{ \cos }\left( {x,\bar{x}} \right)} & {{ \cos }\left( {x,\bar{y}} \right)} & {{ \cos }\left( {x,\bar{z}} \right)} \\ {{ \cos }\left( {y,\bar{x}} \right)} & {{ \cos }\left( {y,\bar{y}} \right)} & {{ \cos }\left( {y,\bar{z}} \right)} \\ {{ \cos }\left( {z,\bar{x}} \right)} & {{ \cos }\left( {z,\bar{y}} \right)} & {{ \cos }\left( {z,\bar{z}} \right)} \\ \end{array} } \right).$$

(16)

Apparently, *T*
^{e} is an orthogonal matrix which permits the stiffness matrix of an element in the global coordinates to be computed as

$$\bar{\varvec{K}}_{e} = \varvec{T}^{{e{\text{T}}}} \varvec{K}_{e} \varvec{T}^{e} .$$

(17)

### 3.2 Strong Coupling of the Elements

Two cases of coupling, “beam to beam” and “beam to shell”, are discussed in this section. Due to the endpoint interpolation, i.e. *C*(−1) = *P*
_{1}, *C*(1) = *P*
_{
n
}, of the beam curves based on NURBS and the coincide exactly with curvature between the beam curve and the connected reflector surface shell, the strong coupling method is suitable for the IGA-based surface error analysis.

#### 3.2.1 Beam to beam coupling

Beams join to each other with a *C*
^{0}-continuous connection, the angle *α* between the beams is assumed unchangeable in the deformed configuration.

As shown in Fig. 4, \(P_{i}^{\gamma }\) denotes the *i*th control point of *γ*th beam. The angle can be described by using the scalar product formula

$$\alpha = \arccos \left( {\frac{{\left( {P_{n}^{1} - P_{n - 1}^{1} } \right)\left( {P_{2}^{2} - P_{n}^{1} } \right)}}{{\left| {P_{n}^{1} - P_{n - 1}^{1} } \right|\left| {P_{2}^{2} - P_{n}^{1} } \right|}}} \right).$$

(18)

KIENDL, et al [38], proposed a bending strip method in which strips of fictitious material with unidirectional bending stiffness and zero membrane stiffness are added at patch interfaces to maintain the angle constraint. The method is efficient, simple to implement, and is applied to the coupling of “beam to beam” in this paper.

#### 3.2.2 Beam to shell coupling

There are mainly two types of the “beam to shell” connection in geometrically, intersection and tangency, as shown in Fig. 5. The latter one is the only type used in the surface error analysis.

As shown in Fig. 6, the beam curve is equivalent to a curve on the surface shell, it’s convenient to make the control points of the beam curve coincident with that of the shell by modifying the surface. The constraint function can be described as

$$\varvec{u}_{a}^{C} = \varvec{u}_{a}^{S} ,$$

(19)

where the superscript *C* and *S* denote the displacement of the *a*th point of beam curve and surface respectively.

### 3.3 RMS error analysis based on IGA shells

The IGA shell element is geometrically exact while the

Langrage element is an approximation of the geometry as shown in Fig. 7, the red dots denote the nodes of the Langrage element. As a result, only the nodes are available to calculate the RMS error as described in Eq. 5 since the point *G*
^{L} in the element have inherent discretization errors. The point *G*
^{I} in the IGA shell element, however, is considered the exact point on the surface. Thus, the arbitrary point in the IGA element is available for the calculation of the RMS error following the Eq. 4.

The four vertices of the IGA shell element, i.e. *ξ* = (−1, 1), (−1, 1), (1, −1), (1, 1), are adopted to determine the unknowns of the best-fit surface by the least square method. The normal deviation of the arbitrary point on the surface can be written as

$$\Delta_{i} \left(\varvec{\xi}\right) = Y\left( {u_{i} \left(\varvec{\xi}\right),v_{i} \left(\varvec{\xi}\right),w_{i} \left(\varvec{\xi}\right)} \right).$$

(20)

The RMS error can then be described as

$$\varepsilon_{{\text{rms}}} = \sqrt {\frac{{\sum\nolimits_{e} {\left[ {\int_{e} {\frac{{\left[ {Y\left( {u_{i}^{e} \left(\varvec{\xi}\right),v_{i}^{e} \left(\varvec{\xi}\right),w_{i}^{e} \left(\varvec{\xi}\right)} \right)} \right]^{2} }}{{1 + \left( {\frac{{r_{i}^{e} \left(\varvec{\xi}\right)}}{2f}} \right)^{2} }}\text{d}A} } \right]} }}{A}} .$$

(21)

The Gauss quadrature is adopted to solve the equation

$$\varepsilon_{{\text{rms}}} = \sqrt {\frac{{\sum\nolimits_{e} {\sum\nolimits_{j = 1}^{gp} {Q^{e} \left( {\varvec{\xi}_{j} } \right)\kappa_{j} } } }}{A}} ,$$

(22)

where \({\kern 1pt} Q^{e} \left( {\varvec{\xi}_{j} } \right) = \frac{{\left[ {Y\left( {u_{i}^{e} \left( {\varvec{\xi}_{j} } \right),v_{i}^{e} \left( {\varvec{\xi}_{j} } \right),w_{i}^{e} \left( {\varvec{\xi}_{j} } \right)} \right)} \right]^{2} }}{{1 + \left( {\frac{{r_{i}^{e} \left( {\varvec{\xi}_{j} } \right)}}{2f}} \right)^{2} }},\)

*κ*
_{
j
} is the weight of the gauss point. The equation is similar with Eq. 5, but however they are different in essence.