Antonio Carminelli

PB-Spline Hybrid Surface Fitting Technique

This work considers the fitting of data points organized in a rectangular array to parametric spline surfaces. Point Based (PB) splines, a generalization of tensor product splines, are adopted. The basic idea of this paper is to fit large scale data with a tensorial B-spline surface and to refine the surface until a specified tolerance is met. Since some isolated domains exceeding tolerance may result, detail features on these domains are modeled by a tensorial B-spline basis with a finer resolution, superimposed by employing the PB-spline approach. The present method leads to an efficient model of free form surfaces, since both large scale data and local geometrical details can be efficiently fitted. Two application examples are presented. The first one concerns the fitting of a set of data points sampled from an interior car trim with a central geometrical detail. The second one refers to the modification of the tensorial B-spline surface representation of a mould in order to create a local adjustment. Considerations regarding strengths and limits of the approach then follow.Copyright

Free Vibration Analysis of Double Curvature Thin Walled Structures by a B-Spline Finite Element Approach

This paper presents a free vibration analysis of general double curvature shell structures using B-spline shape functions and a refinement technique. The shell formulation is developed following the well known Ahmad degenerate approach including the effect of the shear deformation. The formulation is not isoparametric, as a consequence the assumed displacement field is described through non-uniform B-spline functions of any degree. A solution refinement technique is considered by means of a high continuity p-method approach. The eigensolution of a plate, and of single and double curvature shells are obtained by numerical simulation to test the performance of the approach. Solutions are compared with other available analytical and numerical solutions, and discussion follows.Copyright

B-spline Shell Finite Element Updating by Means of Vibration Measurements

Within the context of structural dynamics, Finite Element (FE) models are commonly used to predict the system response. Theoretically derived mathematical models may often be inaccurate, in particular when dealing with complex structures. Several papers on FE models based on B-spline shape functions have been published in recent years (Kagan & Fischer, 2000; Hughes et al, 2005). Some papers showed the superior accuracy of B-spline FE models compared with classic polynomial FE models, especially when dealing with vibration problems (Hughes et al, 2009). This result may be useful in applications such as FE updating. Estimated data from measurements on a real system, such as frequency response functions (FRFs) or modal parameters, can be used to update the FE model. Although there are many papers in the literature dealing with FE updating, several open problems still exist. Updating techniques employing modal data require a previous identification process that can introduce errors, exceeding the level of accuracy required to update FE models (D’ambrogio & Fregolent, 2000). The number of modal parameters employed can usually be smaller than that of the parameters involved in the updating process, resulting in ill-defined formulations that require the use of regularization methods (Friswell et al., 2001; Zapico et al.,2003). Moreover, correlations of analytical and experimental modes are commonly needed for mode shapes pairing. Compared with updating methods using modal parameters as input, methods using FRFs as input present several advantages (Esfandiari et al., 2009; Lin & Zhu, 2006), since several frequency data are available to set an over-determined system of equations, and no correlation analysis for mode pairing is necessary in general. Nevertheless there are some issues concerning the use of FRF residues, such as the number of measurement degrees of freedom (dofs), the selection of frequency data and the ill-conditioning of the resulting system of equations. In addition, common to many FRF updating techniques is the incompatibility between the measurement dofs and the FE model dofs. Such incompatibility is usually considered from a dof number point of view only, measured dofs being a subset of the FE dofs. Reduction or expansion techniques are a common way to treat this kind of incompatibility (Friswell & Mottershead, 1995). A more general approach should also take into account the adoption of different dofs in the two models. As a matter of result, the adoption of B-spline functions as shape functions in a FE

PB-Spline Finite Element Shell Modeling and Refinement Technique

This paper presents a Point Based (PB) spline degenerate shell finite element model to analyze the behavior of thin and moderately thick-walled structures. Complex shapes are modeled with several B-spline patches assembled as in conventional finite element technique. The refinement of the solution is carried out by superimposing a tensorial set of B-spline functions on a patch and employing the PB-spline generalization. The domains for the numerical integration are defined by making use of the retained tensorial framework. Some numerical examples are presented. Considerations regarding strengths and limits of the approach then follow.Copyright

B-Spline Finite Element Formulation for Laminated Composite Shells

This paper presents a finite element formulation for the dynamical analysis of general double curvature laminated composite shell components, commonly used in many engineering applications. The Equivalent Single Layer theory (ESL) was successfully used to predict the dynamical response of composite laminate plates and shells. It is well known that the classic shell theory may not be effective to predict the deformational behavior with sufficient accuracy when dealing with composite shells. The effect of transverse shear deformation should be taken into account. In this paper a first order shear deformation ESL laminated shell model, adopting B-spline functions as approximation functions, is proposed and discussed. The geometry of the shell is described by means of the tensor product of B-spline functions. The displacement field is described by means of tensor product of B-spline shape functions with a different order and number of degrees of freedom with respect to the same formulation used in geometry description, resulting in a non-isoparametric formulation. A solution refinement method, making it possible to increase the order of the displacement shape functions without using the well known B-spline “degree elevation” algorithm, is also proposed. The locking effect was reduced by employing a low-order integration technique. To test the performance of the approach, the static solution of a single curvature shell and the eigensolutions of composite plates were obtained by numerical simulation and are then compared with known solutions. Discussion follows.Copyright

Dynamical Analysis of Shell Structures by Means of B-Spline Shape Functions

This paper presents a free vibration analysis of double curvature free form shaped shell structures using the B-spline shape functions approximation method. It is based on the Ritz method. The shell formulation is developed following the well known Ahmad degenerate approach including the effect of shear deformation. The assumed displacement field is described through non-uniform B-spline functions of any degree. The effect of locking is investigated and both reduced and modified quadrature integration rules are considered with the purpose of increasing the solution accuracy and diminishing the computational cost. Numerical simulation is reported for the evaluation of the eigensolution of plates, and of single and double curvature shells to test the effectiveness and the efficiency of the approach. The presence of spurious zero energy modes both at local and global level was investigated. The solutions are compared with other available analytical and numerical solutions, and discussed in detail.© 2007 ASME

A Local Refinement Technique for B-Spline Finite Element Shell Modeling

This paper presents a refinement technique for a B2-spline degenerate isoparametric shell finite element model for the analysis of the vibrational behavior of thin and moderately thick-walled structures. Complex structures to be refined are modeled by means of FE B-spline patches assembled with C0 continuity as usual in FE technique. The model refinement was performed by adding, on the domain of the selected patch, a tensorial set of polynomial B-spline functions, defined on local clamped knot vectors, and normalizing all the functions so that the resulting displacement field remain polynomial and continuous overall the domain except on the boundaries of the refined subdomain. A degrees of freedom trasformation, based on the knot-insertion algorthim, is adopted in order to guarantee the C0 continuity of the displacement field on the boundaries of the refined subdomain. Two numerical examples are presented in order to test the proposed approach. The natural frequencies of two structures, computed by means of the proposed modelling technique, are compared with reference results available in the literature or computed by means of reference standard FE models. Strengths and limits of the approach are finally discussed.Copyright

B-spline laminate shell finite element updating by means of FRF measurements

A B-spline based FE model updating procedure is proposed. The approach is based on the least squares minimization of an objective function dealing with residues, defined as the difference between the model based response and the experimental measured response, at the same frequency. The B-spline FE model can make it possible to limit the number of dofs. A constraint model and a damping model are proposed, being parameterized by means of B-spline functions. The incompatibility between the measurement dofs and the model dofs is also taken into consideration. An example dealing with a composite car front spoiler is reported, considering the presence of random noise. Results are critically discussed.

Curve and Surface Fitting by Means of Rational B-Spline Functions

This paper presents a method to obtain the mathematical model of a free-form curve or a surface fitting a set of point coordinates by a rational B-spline (NURBS) formulation in the homogeneous R 4 space. A method to evaluate the control points R 4 coordinates is proposed by means of a two step process. In the first step, NURBS weights are evaluated by means of an optimization procedure making it possible to evaluate the best fitting parameterization as well. In the second step, the control point coordinates are computed by means of a linear least squares approach.Copyright