Combined asymptotic finite-element modeling of thin layers for scalar elliptic problems

Thin layers with material properties which differ significantly from those of the adjacent media appear in a variety of applications, as in the form of fiber coatings in composite materials. Fully modeling of such thin layers by standard finite element (FE) analysis is often associated with difficult meshing and high computational cost. Asymptotic procedures which model such thin domains by an interface of no thickness on which appropriate interface conditions are devised have been known in the literature for some time. The present paper shows how the first-order asymptotic interface model proposed by Bövik in 1994, and later generalized by Benveniste, can be incorporated in a FE formulation, to yield an accurate and efficient computational scheme for problems involving thin layers. This is done here for linear scalar elliptic problems in two dimensions, prototyped by steady-state heat conduction. Moreover, it is shown that by somewhat modifying the formulation of the Bövik–Benveniste asymptotic model, the proposed formulation is made to preserve the self-adjointness of the original three-phase problem, thus leading to a symmetric FE stiffness matrix. Numerical examples are presented that demonstrate the performance of the method, and show that the proposed scheme is more cost-effective than the full standard FE modeling of the layer.     

Size-dependent optimal microstructure design based on couple-stress theory

The purpose of this paper is to propose a size-dependent topology optimization formulation of periodic cellular material microstructures, based on the effective couple-stress continuum model. The present formulation consists of finding the optimal layout of material that minimizes the mean compliance of the macrostructure subject to the constraint of permitted material volume fraction. We determine the effective macroscopic couple-stress constitutive constants by analyzing a unit cell with specified boundary conditions with the representative volume element (RVE) method, based on equivalence of strain energy. The computational model is established by the finite element (FE) method, and the design density and FE stiffness of the RVE are related by the solid isotropic material with penalization power (SIMP) law. The required sensitivity formulation for gradient-based optimization algorithm is also derived. Numerical examples demonstrate that this present formulation can express the size effect during the optimization procedure and provide precise topologies without increase in computational cost.     

A 3D Finite Element Method for Flexible Multibody Systems

An efficient finite element (FE) formulation for the simulation of multibody systems is derived from Hamilton's principle. According to the classical assumptions of multibody systems, a large rotation formulation has been chosen, where large rotations and large displacements, but only small deformations of the single bodies are taken into account. The strain tensor is linearized with respect to a co-rotated frame. The present approach uses absolute coordinates for the degrees of freedom and forms an alternative to the floating frame of reference formulation that is based on relative coordinates and describes deformation with respect to a co-rotated frame. Due to the modified strain tensor, the present formulation distinguishes significantly from standard nodal based nonlinear FE methods. Constraints are defined in integral form for every pair of surfaces of two bodies. This leads to a small number of constraint equations and avoids artificial stress singularities. The resulting mass and stiffness matrices are constant apart from a transformation based on a single rotation matrix for each body. The particular structure of this transformation allows to prevent from the usually expensive factorization of the system Jacobian within implicit time--integration methods. The present method has been implemented and tested with the FE-package NGSolve and specific 3D examples are verified with a standard beam formulation.     

A computational procedure for response statistics-based optimization of stochastic non-linear FE-models

An approach for an efficient solution of response statistics-based optimization problems of non-linear FE systems under stochastic loading is presented. A sequential approximate optimization approach, where approximate stochastic analyses are used during portions of the optimization process, is implemented in the proposed formulation. In this approach, analytical approximations of the performance functions in terms of the design variables are considered during the optimization process. The analytical approximations are constructed by combining a mixed linearization approach with a stochastic response sensitivity analysis. The state of the system is defined in terms of the statistical second-moment characteristics of the structural response. The stochastic loading and the response of the system are represented by an orthogonal series expansion of the corresponding covariance matrices. In particular, a truncated Karhunen-Loève (K-L) expansion is applied. The system of non-linear equations is replaced by a statistical equivalent linear system. The evaluation of the K-L vectors is carried out by an efficient procedure that combines local linearization, modal analysis and static response of higher structural modes. An illustrative example is presented that shows the efficiency of the proposed methodology: it considers a building finite element model enforced with non-linear hysteretic devices and subject to a stochastic ground acceleration. Two types of problems are considered: a minimum structural weight design problem and an optimal non-linear device design problem.     

Efficient finite element modelling of reinforced concrete beams retrofitted with fibre reinforced polymers

This paper presents a new simple and efficient two-dimensional frame finite element (FE) able to accurately estimate the load-carrying capacity of reinforced concrete (RC) beams flexurally strengthened with externally bonded fibre reinforced polymer (FRP) strips and plates. The proposed FE, denoted as FRP–FB-beam, considers distributed plasticity with layer-discretization of the cross-sections in the context of a force-based (FB) formulation. The FRP–FB-beam element is able to model collapse due to concrete crushing, reinforcing steel yielding, FRP rupture and FRP debonding.The FRP–FB-beam is used to predict the load-carrying capacity and the applied load-midspan deflection response of RC beams subjected to three- and four-point bending loading. Numerical simulations and experimental measurements are compared based on numerous tests available in the literature and published by different authors. The numerically simulated responses agree remarkably well with the corresponding experimental results. The major features of this frame FE are its simplicity, computational efficiency and weak requirements in terms of FE mesh refinement. These useful features are obtained together with accuracy in the response simulation comparable to more complex, advanced and computationally expensive FEs. Thus, the FRP–FB-beam is suitable for efficient and accurate modelling and analysis of flexural strengthening of RC frame structures with externally bonded FRP sheets/plates and for practical use in design-oriented parametric studies.     

Dynamic delamination modelling using interface elements

Existing techniques in explicit dynamic Finite Element (FE) codes for the analysis of delamination in composite structures and components can be simplistic, using simple stress-based failure function to initiate and propagate delaminations.This paper presents an interface modelling technique for explicit FE codes. The formulation is based on damage mechanics and uses only two constants for each delamination mode; firstly, a stress threshold for damage to commence, and secondly, the critical energy release rate for the particular delamination mode. The model has been implemented into the LLNL DYNA3D Finite Element (FE) code and the LS-DYNA3D commercial FE code.The interface element modelling technique is applied to a series of common fracture toughness based delamination problems, namely the DCB, ENF and MMB tests. The tests are modelled using a simple dynamic relaxation technique, and serves to validate the methodology before application to more complex problems.Explicit Finite Elements codes, such as DYNA3D, are commonly used to solve impact type problems. A modified BOEING impact test at two energy levels is used to illustrate the application of the interface element technique, and it’s coupling to existing in-plane failure models. Simulations are also performed without interface elements to demonstrate the need to include the interface when modelling impact on composite components.     

Hybrid-interface element for thick laminated composite plates

A novel 27-node three-dimensional hexahedral hybrid-interface finite element (FE) model has been presented to analyze laminated composite plates and sandwich plates using the minimum potential energy principle. Fundamental elasticity relationship between components of stress, strain and displacement fields are maintained throughout the elastic continuum as the transverse stress components have been invoked as nodal degrees of freedom. Continuity of the transverse stresses at lamina interface has been maintained. Each lamina is modeled by using hybrid-interface elements at the top and the bottom interfaces and conventional displacement based elements sandwiched between these interfaces. Results obtained from the present formulation have found to be in excellent agreement with the elasticity solutions for thin and thick composite cross-ply, angle-ply laminates, as well as sandwich plates. Additional results have also been presented on the variation of the transverse strains to highlight magnitude of discontinuity in these quantities due to difference in properties of face and core materials of sandwich plates. Present formulation can be used effectively to interface hybrid formulation that uses transverse stresses and displacements as degrees of freedom with conventional purely displacement based formulation for realistic estimates of the transverse stresses.     

Application of a three-field nonlinear fluid-structure formulation to the prediction of the aeroelastic parameters of an F-16 fighter

We overview a three-field formulation of coupled fluid-structure interaction problems where the flow is modeled by the arbitrary Lagrangian-Eulerian form of either the Euler or Navier-Stokes equations, the structure is represented by a detailed finite element (FE) model, and the fluid grid is unstructured, dynamic, and constructed by a robust structure analogy method. We discuss the latest advances in the computational algorithms associated with this approach for modeling aeroelastic problems. We apply the three-field nonlinear computational framework to the prediction of the aeroelastic frequencies and damping coefficients of an F-16 configuration in various subsonic, transonic, and supersonic regimes. We consider for this purpose both the popular two-dimensional typical wing section model and a detailed three-dimensional FE model of the structure, and compare in both cases the obtained numerical results with flight test data. We comment on the advantages and shortfalls of both approaches, and on the feasibility as well as the merit of the three-field formulation of nonlinear aeroelasticity for the extraction of flutter envelopes.     

Evaluation of three lattice Boltzmann models for multiphase flows in porous media

A free energy (FE) model, the Shan–Chen (S–C) model, and the Rothman and Keller (R–K) model are studied numerically to evaluate their performance in modeling two-dimensional (2D) immiscible two-phase flow in porous media on the pore scale. The FE model is proved to satisfy the Galilean invariance through a numerical test and the mass conservation of each component in the simulations is exact. Two-phase layered flow in a channel with different viscosity ratios was simulated. Comparing with analytical solutions, we see that the FE model and the R–K model can give very accurate results for flows with large viscosity ratios. In terms of accuracy and stability, the FE model and the R–K model are much better than the S–C model. Co-current and countercurrent two-phase flows in complex homogeneous media were simulated and the relative permeabilities were obtained. Again, it is found that the FE model is as good as the R–K model in terms of accuracy and efficiency. The FE model is shown to be a good tool for the study of two-phase flows with high viscosity ratios in porous media.     

Comparison of different formulations of 2D beam elements based on Bond Graph technique

Bond Graphs are well suited for modelling multibody systems. In this paper modelling of planar flexible beams undergoing large overall motions are studied based on finite element (FE) technique. Two well-known approaches are used – the co-rotational (CR) and absolute nodal coordinate (ANC) formulation. Two ANC formulations are analyzed – one in which elastic forces is described using classical beam theory in a local coordinate frame, and another based on a global continuum mechanics approach. Starting from these classical formulations velocity formulations are developed and used to develop Bond Graph FE components. The effect of gravity has been considered as well. These components can be put in libraries and used for systematic Bond Graph flexible body model development. It is shown that Bond Graph technique is capable of dealing with different flexible body formulations and can be used as a general approach in parallel to other modelling approaches. Models are developed and simulations are performed using the object oriented environment of BondSim. Owing to the object oriented approach, transformation from one to the other model is relatively simply. The results are illustrated by suitable examples and they confirm accuracy of the developed models. It was shown that the CR approach offers much better performance than the both ANC formulations.     

The use of finite-element and boundary-element models for predicting the vibro-acoustic behaviour of layered structures

Layered structures made from elastic and porous materials are widely used as insulation systems in the automotive industry. They have a complex dynamic behaviour that is influenced by the various interaction mechanisms within the porous material. This paper concentrates on modelling these systems using a three-dimensional finite-element (FE) approach for the structure combined with a boundary-element (BE) procedure for the acoustic radiation process. The key part is a Biot model for the two-phase porous material. A mixed displacement formulation is selected. The model can be used for predicting the surface impedance and/or the transmission loss characteristics of layered material. The combined use of this FE/BE model enables evaluation of the acoustic response (radiated power, field pressure). Numerical applications are presented in order to show the capabilities of the developed procedures.     

Applications of hybrid optimization techniques for model updating of rotor shafts

This paper presents an approach by combining the genetic algorithm (GA) with simulated annealing (SA) algorithm for enhancing finite element (FE) model updating. The proposed algorithm has been applied to two typical rotor shafts to test the superiority of the technique. It also gives a detailed comparison of the natural frequencies and frequency response functions (FRFs) obtained from experimental modal testing, the initial FE model and FE models updated by GA, SA, and combination of GA and SA (GA–SA). The results concluded that the GA, SA, and GA–SA are powerful optimization techniques which can be successfully applied to FE model updating, but the appropriate choice of the updating parameters and objective function is of great importance in the iterative process. Generally, the natural frequencies and FRFs obtained from FE model updated by GA–SA show the best agreement with experiments than those obtained from the initial FE model and FE models updated by GA and SA independently.     

Obtaining a hyperelastic non-linear orthotropic material model via inverse bubble inflation analysis

An inverse bubble inflation test is proposed utilizing full displacement field matching to obtain non-linear material models suitable for the Finite Element (FE) method. In this paper a known non-linear orthotropic material model is assumed as the solution for the inverse method to illustrate the process. A bubble inflation FE analysis is performed with the known material model to determine the load and displacement field from the assumed material. Polynomial surfaces are fit to the nodal displacement values of the FE model, such that the entire displacement field is stored as three unique polynomial surfaces. An error formulation was established to quantify the quality of fit between different bubble inflation displacement fields. Gradient based optimization is used to obtain the assumed material model by matching the full displacement field. The inverse bubble inflation test successful produces a non-linear orthotropic model that is analogous to the assumed non-linear orthotropic material, and thus demonstrates that the inverse bubble inflation analysis would be able to characterize other non-linear orthotropic materials.     

Arbitrary active constrained layer damping treatments on beams: Finite element modelling and experimental validation

This paper concerns the analytical formulation and finite element modelling of arbitrary active constrained layer damping (ACLD) treatments applied to beams. A partial layerwise theory is utilized to define the displacement field of beams with an arbitrary number of elastic, viscoelastic and piezoelectric layers attached to both surfaces, and a fully coupled electro-mechanical theory is considered for modelling the behavior of the piezoelectric layers. The damping of the viscoelastic layers is modelled by the complex modulus approach. The weak forms of the analytical formulation, governing the motion and electric charge equilibrium, are presented. Based on the weak forms, a one-dimensional finite element (FE) model is developed, with the nodal mechanical degrees of freedom being the axial displacement, transverse displacement and the rotation of the mid-plane of the host beam and the rotations of the individual layers, and the electrical elemental degrees of freedom being the electrical potential difference of each piezoelectric layer. Frequency response functions were measured experimentally and evaluated numerically for a freely suspended aluminium beam with an ACLD patch. In order to validate the FE model the results are presented and discussed.     

Nonlinear tracking in a diffusion process with a Bayesian filter and the finite element method

A new approach to nonlinear state estimation and object tracking from indirect observations of a continuous time process is examined. Stochastic differential equations (SDEs) are employed to model the dynamics of the unobservable state. Tracking problems in the plane subject to boundaries on the state-space do not in general provide analytical solutions. A widely used numerical approach is the sequential Monte Carlo (SMC) method which relies on stochastic simulations to approximate state densities. For off-line analysis, however, accurate smoothed state density and parameter estimation can become complicated using SMC because Monte Carlo randomness is introduced. The finite element (FE) method solves the Kolmogorov equations of the SDE numerically on a triangular unstructured mesh for which boundary conditions to the state-space are simple to incorporate. The FE approach to nonlinear state estimation is suited for off-line data analysis because the computed smoothed state densities, maximum a posteriori parameter estimates and state sequence are deterministic conditional on the finite element mesh and the observations. The proposed method is conceptually similar to existing point-mass filtering methods, but is computationally more advanced and generally applicable. The performance of the FE estimators in relation to SMC and to the resolution of the spatial discretization is examined empirically through simulation. A real-data case study involving fish tracking is also analysed.     

A population dynamics model to describe gene frequencies in evolutionary algorithms

The performance of evolutionary algorithms (EAs) may heavily depend severely on a suitable choice of parameters such as mutation and crossover rates. Several methods to adjust those parameters have been developed in order to enhance EA performance. For this purpose, it is important to understand the EA dynamics, i.e., to appreciate the behavior of the population. Hence, this paper presents a new model of population dynamics to describe and predict the diversity in any particular generation. The formulation is based on selecting the probability density function of each individual. The population dynamics proposed is modeled for a generational population. The model was tested in several case studies of different population sizes. The results suggest that the prediction error decreases as the population size increases.     

Application of asymptotic waveform approximation technique to hybrid FE/BI method for 3D scattering

The asymptotic waveform evaluation (AWE) technique is a rational function ap- proximation method in computational mathematics, which is used in many appli- cations in computational electromagnetics. In this paper, the performance of the AWE technique in conjunction with hybrid finite element/boundary integral (FE/BI) method is firstly investigated. The formulation of the AWE applied in hybrid FE/BI method is given in detail. The characteristic implementation of the application of the AWE to the hybrid FE/BI method is discussed. Numerical results demonstrate that the AWE technique can greatly speed up the hybrid FE/BI method to acquire wide-band and wide-angle backscatter radar-cross-section (RCS) by complex tar- gets.     

On the potential applications of a 3D random finite element model for the simulation of shot peening

Shot peening is a cold-working process that is used mainly to improve the fatigue life of metallic components. Experimental investigation of the mechanisms involved in shot peening is very expensive and complicated. Therefore, the Finite Element (FE) method has been recognized as an effective mean for characterizing the shot peening process and several types of FE models have been developed to evaluate the effects of shot peening parameters. However, in most of the existing FE models, the shot peening sequence and impact location were defined a priori. It is therefore the purpose of this study to consider the random property of the shot peening process. A novel 3D FE model with multiple randomly distributed shots was developed combining a Matlab program with the ANSYS preprocessor. The explicit solver LS-DYNA has been used to simulate the dynamic impingement process. Several potential applications of this novel model such as: the quantitative relationship of the peening intensity, coverage and roughness with respect to the number of shots have been presented. Moreover, simulations with multiple oblique impacts have been carried out in order to compare with results from normal impingements. Our work shows that such a computing strategy can help understanding and predicting the shot peening results better than conventional FE simulations.     

Procreated C displacement generic plate vibration model

Procreation of the C¹ continuous displacement FE efficient vibration model, which can be attained by exploiting the unparalleled conciseness and cost-objectiveness of a computer automated exact analytical integrator, is considered. The proposed Kirchhoff-conforming, rectangular four dof/node plate FE vibration simulator is demonstrated to achieve high accuracy with low CPU expenditure in a straightforward and systematic framework without calling for any ad hoc mechanisms. Due to idiosyncrasies of the formulation, the code has the potential for providing high-speed computational power to aid accurate and reliable identification of a wide spectrum of vibration modes that faithfully captures the physical behavior of thin plates, frequently encountered in real-life engineering applications.     

Development of parallel explicit finite element sheet forming simulation system based on GPU architecture

Sheet forming simulation is very important for vehicle body design. Due to the increase of complexity and scale of the CAE model, a tradeoff between the accuracy and efficiency become the bottleneck for application. Therefore, a parallel explicit finite element (FE) based on graphics processing unit (GPU) architecture for sheet forming is developed. Implementation details with computer unified device architecture (CUDA) are considered in this work. A pre-index strategy is suggested for parallelization of nodal force assembling. Parallel reduction method is introduced to calculation of the global time step. To ensure the reliability and accuracy of the GPU-based program, double precision floating and intrinsic functions are implemented for the explicit FE computing. The simulation results based on a commercial NVIDIA GTX285 device can obtain about 27X speedup than on a Intel Q8200 CPU, which demonstrates the efficiency of the parallel sheet forming simulation system.