F. Trochu

Generation of a finite element MESH from stereolithography (STL) files

Abstract The aim of the method proposed here is to show the possibility of generating adaptive surface meshes suitable for the finite element method, directly from an approximated boundary representation of an object created with CAD software. First, we describe the boundary representation, which is composed of a simple triangulation of the surface of the object. Then we will show how to obtain a conforming size-adapted mesh. The size adaptation is made considering geometrical approximation and with respect to an isotropic size map provided by an error estimator. The mesh can be used “as is” for a finite element computation (with shell elements), or can be used as a surface mesh to initiate a volume meshing algorithm (Delaunay or advancing front). The principle used to generate the mesh is based on the Delaunay method, which is associated with refinement algorithms, and smoothing. Finally, we will show that not using the parametric representation of the geometrical model allows us to override some of the limitations of conventional meshing software that is based on an exact representation of the geometry.

Thermomechanical Properties during Cure of Glass-Polyester RTM Composites: Elastic and Viscoelastic Modeling

Resin transfer molding (RTM) is a widely used technique for the manufacturing of composite parts. A proper selection of process parameters is the key to yield successful molding results and obtain a good part. During composite consolidation, resin cure, also called chemical conversion, plays a decisive role on the final mechanical properties of the part. The modeling of resin kinetics and the evolution of composite properties during cure are crucial for process optimization. In this paper, the curing of a thermosetting polyester resin is studied by differential scanning calorimetry (DSC). A semiempirical autocatalytic model is developed to describe the kinetics of the chemical reaction. The model accounts for the maximum degree of polymerization as a function of cure temperature and induction time, i.e., the time required to attain total inhibitor degradation. The evolution of mechanical properties during resin cure for two glass-polyester composites is also studied with a dynamical mechanical thermal analyzer (DMTA) and a thermomechanical analyzer (TMA). Given that for a low chemical conversion, the elastic properties of the resin remain low, an initial degree of polymerization called after gel point (AGP) is introduced in the analysis of the mechanical properties during cure. A normalized elastic modulus is defined from the value at AGP, taken as a reference. The normalized elastic modulus is then compared to the polymerization degree. For pure resin samples, the logarithm of chemical conversion is found to be almost linearly related to the logarithm of the elastic modulus. Based on this comparison, a thermochemical model is proposed to describe the evolution of mechanical properties during the cure of composite samples with different fiber volume fractions. The viscoelastic behavior is also determined by performing stress relaxation tests with the DMTA. Resin specimens are tested for different cure states below the glass transition temperature, and master curves of stress relaxation during cure are constructed by applying the time-temperature superposition principle. The measurements depict the relaxation modulus of polyester resins as sharply affected by the degree of polymerization. Based on the experimental data, a relaxation modulus is modeled in a thermorheologically simple manner using exponential and power laws. Finally, a linear volume change model is constructed based on the TMA measurements of thermal expansion and resin shrinkage. The volume changes resulting from composite expansion-contraction and resin polymerization shrinkage are modeled as a function of temperature and degree of polymerization. The purpose of this work is to develop appropriate models of chemo- and thermomechanical behaviors of glass-polyester composites during cure. A resin cure kinetics model is developed by adding the glass transition effects to the J.L.B. model. For the mechanical properties, two new models are presented to account for the elastic and viscoelastic behaviors of the resin and the composite. Finally, the coefficients of the volume changes model are measured to account for the composite thermal expansion-contraction and resin chemical shrinkage. These models will be used in future investigations for thermal and curing optimization of composites processed by resin transfer molding.

Limitations of a Boundary-Fitted Finite Difference Method for the Simulation of the Resin Transfer Molding Process

Resin Transfer Molding (RTM) is a commonly used fabrication method for large parts of fiber reinforced composites. The reinforcement consists of fiber mats or woven rovings first laid inside a mold cavity, then catalyzed resin is injected through prop erly positioned injection ports. A numerical model based on Darcys law has been devel oped. It permits to simulate, by a boundary-fitted finite difference method, the filling of two-dimensional molds of arbitrary shapes. The resin pressure distribution and the resin front positions can be obtained at each time step. Calculated results on selected mold geometries are compared with the experimental observations and discussed with those ob tained by other investigators. Finally the experimental and computational limitations of the proposed simulation are pointed out and illustrated for particular examples.

Geometric algorithms for the intersection of curves and surfaces

Abstract The problem of finding the intersection of curves and surfaces arises in numerous computer aided design applications. The methods generally used rely on iterative numerical techniques based on the solution of a set of non-linear equations. These systems of equations are generally local and need adequate starting points in order to yield convergent solutions. This article presents two general algorithms based on geometric considerations to find the intersections of C0 curves and surfaces. The first method can be applied when one object is defined by a parametric equation and the other by an implicit equation. The second method is based on a succession of orthogonal projections from one object to the other. The same algorithm can be applied to curves and surfaces. These methods are implemented in the general framework provided by dual kriging for parametric curve and surface modelling. Finally, the conjugate tangent approach can speed up considerably the algorithm by considering alternatively tangent lines or planes in the iterative process together with orthogonal projections.

Modeling the edge effect in liquid composites molding

In Liquid Composite Molding (LCM) processes such as RTM or SRIM, preformed fabrics are preplaced in the mold cavity. The mold is then closed and a liquid thermoset resin is injected. Since it is difficult to precisely cut the fiber preform to the exact shape of the mold, sometimes a gap exists between the preform and the mold edge. This gap, even small (1 or 2 mm), can create a preferential flow path for the resin which disrupts the filling of the mold cavity. Such flow perturbation is called the edge effect. With existing numerical simulation models it is possible to simulate an edge effect by locally changing the permeability. However, this is not satisfactory. The ideal case will be a model to predict the edge effect from the geometry of the gap and the porosity of the surrounding material. To respond to this need, this paper presents an analysis of the flow patterns using appropriate flow equations in the open channel and Darcys law in the porous medium. From this an equivalent porous medium is defined for the channel for which an equivalent permeability tensor can be computed. Two geometric models to predict the edge effect are presented. The first model is derived from the Navier-Stokes equation in the channel. In the second model, the flow is assumed to take place in an equivalent cylindrical channel as in Poiseuille flow. However, these models cannot cover all cases. To evaluate the applicability of these simple models, a parameter called the transverse flow factor is defined. For finite element flow simulation an equation to define the equivalent permeability of the first row of elements encompassing the open channel is given. Finally, experimental as well as simulation results are presented.

NONLINEAR FINITE ELEMENT SIMULATION OF SUPERELASTIC SHAPE MEMORY ALLOY PARTS

Abstract This article presents a methodology to simulate the nonlinear thermomechanical behaviour of shape memory alloys (SMA) by the finite element method. After a brief presentation of the remarkable thermomechanical properties of SMA materials, a general and simplified constitutive law is formulated based on plastic flow theory, which takes into account the stress and temperature induced phase transformation in the alloy. The reason for this connection between plasticity and the superelastic behaviour of SMA lies in the phase transformation itself, the effect of which is somehow similar to a plastic flow. This approach, however, differs from plasticity in the sense that upon unloading the initial state of the material can be recovered through a hysteresis cycle. To simulate this hysteresis, two different von Mises plastic criteria for three-dimensional problems are used for the loading and unloading procedures, respectively, so that unloading can also be treated as a transition from elasticity to plasticity. It is first suggested to use a bilinear model as uniaxial material law. Then a more precise model based on dual kriging interpolation is also proposed and implemented. The nonlinear finite element equations of the approximate problem are briefly stated and the methodology is validated by comparison with experimental results. Finally, one example of industrial application is given concerning stress analysis of a SMA spring disc.

Directional Permeability Measurement of Deformed Reinforcement

This paper presents a method for the evaluation of the anisotropic permeability of deformed woven fabrics. In order to conform to complex shapes, the woven fabric must undergo a certain amount of deformation. This deformation disturbed the resin flow and the filling of the mold cavity in liquid molding processes such as resin transfer molding (RTM). It is important for computer simulations of the filling process to predict the change of directional permeabilities k, and k2 caused by the deformation of the fabrics. Several flow experiments were conducted on a nonstitched woven fabric with different deformation angles. The deformation affects the global permeability and the directional permeabilities k1, k2. At the same time, the principal permeability axes are shifted. The resulting permeability is related to the orientation of the fabric. This article presents characteristic permeability curves for the fabric JBMartin NCS 81053-A as a function of the shearing angle or, alternatively, of the resulting fiber fraction. It is possible to conclude that for the fabric tested, the shearing angle of the reinforcement will have a definite influence on the resin flow pattern.

Prediction of fibre orientation and net shape definition of complex composite parts

Prediction of fibre orientation is important to assess the mechanical properties of a composite part. When the part is manufactured by a liquid moulding process, the resin flow through the preform depends also on local fibre orientation. In this article, a methodology based on dual kriging surface interpolation is presented for the prediction of fibre orientation in composite materials. By calculating the coordinates of each node of the fibre network, it is possible to determine the fibre orientation when the fabric is draped on a complex surface and to calculate the angle between two adjacent fibres. The methodology proposed here combines the definition of a parametric surface by dual kriging with a simple algorithm allowing free rotations of the fibres around the nodes of the fabric. Numerical results from simulations confirm the validity of this method to predict fibre orientation. The same approach can be used also for net shape definition of a fabric before draping on a complex surface. However, implementation of this methodology is computer-intensive and the accuracy is limited by round-off errors.

Touch probe radius compensation for coordinate measurement using kriging interpolation

Abstract Obtaining CAD (computer aided design) descriptions of actual parts having complex surfaces is a key part of the process of reverse engineering. This paper is concerned with the estimation of actual surfaces using coordinate measuring machines fitted with a spherically tipped touch probe. In particular, it addresses in detail the problem of probe radius compensation. A general mathematical model, using kriging, is proposed which first generates the initial probe centre surface and then estimates the compensated or part surface. The compensation is achieved using normal vectors to the initial probe centre surface at each measured point to compensate for the probe radius. The method is validated experimentally on known and free-form surfaces.

Simulation of Compression Resin Transfer Molding with Displacement Control

In this paper a mathematical model of two-dimensional resin flow through fiber reinforcements in compression resin transfer molding (CRTM) is presented. The preform is partially filled by resin during the injection phase. Then it is compressed by the mobile upper part of the mold. The resin flow in the fiber bed is governed by Darcys law according to the theory of flows in saturated porous media. The consolidation of the saturated preform is described by the total mass conservation equation. A filling algorithm based on resin conservation on a deformable grid is used to advance the flow front at each time step. Resin pressure and velocity are calculated by the finite element method. The accuracy of the model is verified by evaluation of the resin mass balance, calculation of the resin pressure and progression of the flow front in time. Comparison of predicted results with analytical solutions is also presented.