12 W. Witteveen et al. Brief Outline of the Theory Only the idea of the theory is outlined below. The detailed derivation can be found in [3]. It is assumed that a usable FE model of the structure including the flange area is available. With the use of proper trial vectors (commonly called “modes”), the normal displacement (=gap) between the two contact surfaces can be computed in the form of g =Φgq (1) where the (C×1) vector g holds the normal distance of all node-to-node contact pairs which form the contact area. The (C×Q) matrix Φg holds in its columns the portion of the modes which belong to the gap area and the (Q×1) vector q holds the scaling factors (commonly called “modal coordinates”) of the modes. It is obvious that the mode base must be able to capture global structural deformations like vibrations and very local ones like the deformations in the contact area. This can be ensured by extending proven mode bases with so-called contact modes. This allows both types of deformation to be represented with sufficient accuracy. For more information on contact modes, the interested reader is referred to [4]. Regarding linear strains, the relation ε =Φεq (2) also apply where the (E×1) vector ε holds all potential strain measurements at E positions on the structure. The (E×Q) matrix Φε maps the modal coordinates to the strains. By the use of the pseudo inverse equation (2) can be rearranged and plugged into (1) so that g =Gε (3) with the (C×E)matrix G=ΦgΦε† (4) is obtained. Relation (3) is not useful because it holds all potential strain gauges which is far too much for practical use. Training data is used to reduce the number of strain gauges. We assume that, for example through simulation, T training data ε1 to εT are available. They are arranged column by column in the (E×T) matrix ET. In a next step Proper Orthogonal Decomposition (POD, see exemplarily [5]) is used to extract a S dimensional subspace which optimally approximates all T training data in an Euclidian sense. The S vectors which span the subspace are arranged column wise in a (E×S) matrix ES. Now the Discrete Empirical Interpolation Method (DEIM, see exemplarily [6]) can be used for a somehow optimal extraction of S sensor positions out of E potential ones. The application of DEIM to ES gives a (S×E) matrix BT and a (E×S)matrixEDwith the properties εD =BTε (5) and ε =EDεD. (6) Note, that in relation (5) the (S×E) vector εDholds a Sdimensional subset of the entries inε. Consequently, BT selects the actual used sensors out of all potential ones. The matrix ED in relation (6) is used to computed all potential strains on the base of the selected subset stored inεD. When (6) is plugged into (3) the final relation g =GDεD (7) with the time invariant (C×S)matrix GD =GED (8) is obtained. Relation (7) can be used to compute the gap distributiong based on the strain measurements inεD. Numerical Example The proposed method is tested on a purely numerical example. The aim is to give a kind of numerical prove that the deformation state of a joint, and thus the tightness, can be determined using just a few strain gauge measurements. As shown in Fig. 1, two pipes are connected with a flange. The dimensions and all other relevant parameters can be found in [3]. The two flange rings are connected via 12 bores (M20 screws). The entire structure is rigidly mounted on one side and 3 forces are applied on the other side (Fx, Fy and Fz). The FE model consists of 131040 linear hexahedron elements and 179162 nodes. The contact surfaces of the flange are congruently meshed with 7944 (=C) node pairs. The contact pressures and the friction forces are computed by nonlinear penalty laws which can be found [3].
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