Initial model results used estimations of the modulus of the fiber and matrix using common values found for both materials. The model was refined to match the tensile testing data of the matrix and fiber upon completion of material tests to provide a more representative comparison between the theoretical model and experimental results. 3.3 Direct Analysis The direct analysis is performed using a tightly coupled fluid-structure co-simulation in which the structural problem is solved using the free general purpose multibody dynamics solver MBDyn1 [16] and the fluid problem is solved using a dedicated solver based on FEniCS2 [11, 12] where a collection of Python statements inherit the mathematical structure of the problem and automatically generate low level code. The fluid dynamics code is based on a stabilized finite element approximation of the unsteady Navier-Stokes equations (often referred to in the literature as G2 method [17]). The multibody solver is coupled with the external fluid dynamic code by means of a general-purpose, meshless boundary interfacing approach based on Moving Least Squares with Radial Basis Function, as presented in [14]. This technique allows for an approximation of the field of the structural displacements, and velocities at the aerodynamic interface nodes. The membrane element, implemented in MBDyn as shown in [10], is formulated as a four-node isoparametric element based on second Piola-Kirchhoff type membranal resultants. The classical Enhanced Assumed Strains (EAS) method [18] is exploited to improve the response of the element: seven additional variables for each membrane element are added to the strain vector (see for example [19] for details). The stress tensor in vector form can be expressed as a function of the strain tensor, reorganized in the same manner, using the constitutive law of the membrane element, e.g.: σ11 σ22 σ12 8 < : 9 = ; ¼D ε11 ε22 ε12 8 < : 9 = ; ð3:2Þ where D¼ E 1 v2 1 v 0 v 1 0 0 0 1 v ð Þ=2 2 4 3 5 ð 3:3Þ In case of homogeneous constitutive properties, the forces per unit span are readily obtained by multiplying the stresses by the thickness h of the membrane (otherwise, thickness-wise integration is required). Generically anisotropic constitutive properties can be defined with a positive, symmetric D matrix, arbitrarily set by the user. 3.4 Experimental Data Re-sampling Measurements provided by DIC [5] include: (i) the reference location in space of an arbitrary set of points on the surface, chosen by the DIC algorithm when the measurement system is activated, (ii) the displacements of the corresponding points in the current sample, and (iii) an estimate of the in-plane strains. Data preparation, for both the measured strains and the measured displacements used for correlation, requires re-sampling of unstructured measured fields onto the grid that is subsequently used for the analysis; for this purpose, a meshless mapping procedure is used. The mapping [14] produces a linear interpolation operator H, from the measurement domain (·)m, to the virtual sensing domain (·)v, namely xv ¼Hxm. Operator H is computed based on the initial positions of both domains; from that point on, it is used to map an arbitrary configuration of the measure domain onto the virtual sensing domain. The participation of each component of a measure point’s position to the mapping of the corresponding component of a virtual point is the same, i.e., the mapping is isotropic. As a consequence, any scalar field, as well as each component of any vector field, can be mapped separately using a subset of matrix H, obtained for example by extracting every one out of three columns and rows of matrix H 3 Manufacturing and Characterization of Anisotropic Membranes for Micro Air Vehicles 21
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