Shape Error Analysis of Functional Surface Based on Isogeometrical Approach
© The Author(s) 2017
Received: 4 May 2016
Accepted: 2 April 2017
Published: 13 April 2017
The construction of traditional finite element geometry (i.e., the meshing procedure) is time consuming and creates geometric errors. The drawbacks can be overcame by the Isogeometric Analysis (IGA), which integrates the computer aided design and structural analysis in a unified way. A new IGA beam element is developed by integrating the displacement field of the element, which is approximated by the NURBS basis, with the internal work formula of Euler-Bernoulli beam theory with the small deformation and elastic assumptions. Two cases of the strong coupling of IGA elements, “beam to beam” and “beam to shell”, are also discussed. The maximum relative errors of the deformation in the three directions of cantilever beam benchmark problem between analytical solutions and IGA solutions are less than 0.1%, which illustrate the good performance of the developed IGA beam element. In addition, the application of the developed IGA beam element in the Root Mean Square (RMS) error analysis of reflector antenna surface, which is a kind of typical functional surface whose precision is closely related to the product’s performance, indicates that no matter how coarse the discretization is, the IGA method is able to achieve the accurate solution with less degrees of freedom than standard Finite Element Analysis (FEA). The proposed research provides an effective alternative to standard FEA for shape error analysis of functional surface.
Functional surface is a type of complex surface, which is ubiquity in mechanical and electrical equipment, realizes specific physical performances, such as electric, magnetic, optic and thermodynamic performance. The accuracy of functional surface, which is the critical function structure of a product, has a significant impact on the product’s performance. Antenna surface is a kind of typical functional surface, which realizes transmitting and receiving of electromagnetic signal. As in the case of reflector antenna where the reflector surface is such a functional surface, the influences on reflector surface precision from external loads, manufacturing and assembling errors [1–5] are considerable in changing the amplitude and phase distribute of antenna aperture to affect the far electric field of antenna. And the surface root mean square (RMS) of half-path-length error is always adopted to estimate the gain degradation according to the Ruze equation . In recent years, more and more large reflector antennas are equipped with shape control systems to estimate the surface errors. TANAKA  added intentional deformations on an antenna surface using the surface adjustment mechanisms to estimate the surface errors. And a dynamic shape control strategy of deployable mesh reflectors via feedback approaches was proposed by XIE, et al . Other adjustment strategies have been investigated by several researchers [9–11]. In order to get the control inputs to actively adjust the surface shape, measurement methods and numerical simulations are adopted to estimate the surface error. The phase-retrieval holographic analysis [12, 13], photogrammetry measurements , etc. are widely used to measure shapes of reflector antennas. Although measurement methods could easily acquire the surface displacement distribution, the measure accuracies rely on the special equipment and the antenna’s wavelength. The numerical simulations, Finite Element Analysis(FEA), are good alternates in shape error analysis due to the advantage in optimization of the initial prototype designs. YOON  formulated a shape error minimization problem for a mechanically deformable reflector antenna structure in the frame work of the FEA. YOU, et al , used a 3-nodes laminated shell element based on Lagrange’s equations to study the characteristics of the reflector. DU, et al , employed the FEA to calculate the sensitivity matrix of the nodal displacement to deal with the worst-case optimization problem of cable mesh reflector antenna. However, only the displacements of the elements nodes can be applied to calculate the RMS error since the points in the element have inherent discretization errors which are especially bad for surfaces with a relatively coarse mesh. Refined mesh is required to improve the simulation accuracy, which would in turn makes the calculation more time consuming.
Isogeometric analysis (IGA)  is a method aimed at avoiding the discretization errors by using the same basis for analysis as is used to describe the geometry, thus enabling IGA to discretize the analysis models exactly. The necessary continuities between elements, C 1 continuity for Kirchhoff-Love element as an example, can be easily achieved by the high-order geometric basis functions. The method is now further developed in many areas including structural analysis [19–21], fluid-structure interaction , shape optimization [23, 24], topology optimization [25, 26], electromagnetic analysis , etc.. New IGA elements [28, 29] are developed by researches. However, the non-interpolatory nature of the geometric basis functions makes the imposition of even the simple boundary conditions more difficult. Weak and strong methods are studied to the coupling [30–32] and boundary condition imposition issues [33, 34] to perfect the novel method. The method is now capable to deal with the majority of engineering issues. The exact geometric discretization of IGA enables us to achieve more accurate solutions by much less degrees of freedom than that of traditional FEA.
The paper is organized as follows. In section 2, a brief introduction of the surface error estimation is presented, include the relationship between the RMS half-path-length error and antenna gains, the normal deviation, etc. In section 3, a rotation-free three-dimensional IGA beam element combined with Bézier extraction is developed, and a Kirchhoff-Love shell element is also introduced in brief. Then the coupling of the two elements is presented. Moreover, at the end of the section, the RMS error is written in the form of IGA. In section 4, several examples are presented to verify the effectiveness of the developed beam element and its application in the RMS error analysis of antenna reflector. Finally, concluding remarks are given in section 5.
2 Surface Error Estimation
3 Isogeometric Analysis for Surface Error Analysis
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
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.
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.
KIENDL, et al , 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
3.3 RMS error analysis based on IGA shells
The IGA shell element is geometrically exact while the
κ j is the weight of the gauss point. The equation is similar with Eq. 5, but however they are different in essence.
4 Numerical Examples
In the following, two numerical examples are presented to reveal the overall performance of the three dimensional Euler-Bernoulli Beam IGA element and the application of IGA in RMS error analysis. First, a Cantilever beam subjected to a point force is introduced to verify the accuracy of the beam element with the analytical solution. And then a parabolic antenna modeled by the Kirchhoff-Love shell and Euler-Bernoulli Beam IGA element is prepared for the RMS error analysis.
4.1 Cantilever beam
4.2 Surface error analysis of a reflector antenna
Convergence analysis of max displacement and RMS error
Number of control points N
Max displacement D/mm
RMS error ε rms/mm
A new IGA beam element is developed by integrating the displacement field of the element, which is approximated by the NURBS basis, with the internal work formula of Euler-Bernoulli beam theory with the small deformation and elastic assumptions.
Two cases of coupling, “beam to beam” and “beam to shell”, are discussed. Due to the endpoint interpolation 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.
Due to the geometrically exact no matter how coarse the discretization is and the higher-order basis functions, the IGA method is able to achieve the accurate solution with less degrees of freedom than traditional FEA and the arbitrary point in the IGA element is available for the calculation of the RMS error.
The cantilever beam benchmark problem was chosen to demonstrate the good performance of the developed IGA beam element. The maximum relative errors of the deformation in the three directions between analytical solutions and IGA solutions are less than 0.1%.
An antenna model, which is composed of a main reflector and a bracket, is discretized by the IGA shell and beam elements respectively. By coupling the elements strongly, the IGA method is applied in the functional surface error analysis of the antenna reflector successfully. It is clear that the IGA approach reaches the convergence precision with much less control points than traditional FEA.
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