Formulation Of Finite Element Models Research Articles

Lagrangian Finite Element Model Formulation and Experimental Validation of the Laser Impact Weld Process for Ti/Brass Joining

Information on the flyer deformation during laser impact welding (LIW) is an important aspect to consider when high reliability of the welded components is required. For this reason, accurate numerical models simulating thermal and mechanical aspects are needed. In the present work, the cross-section morphology during LIW of Ti/Brass joints at varying laser pulse energies is modeled by a 2D finite element (FE) model. A hydrodynamic plasma pressure model able to describe the evolution of the pressure load step by step, taking into account the progressive deformation of the flyer, was implemented. Hence, this paper proposes an alternative method to the conventional node concentrated forces or predefined velocity as flyer boundary conditions. The levels of the equivalent plastic strain (PEEQ), shear stress, and critical flyer velocity at the collision point were used as reference parameters to predict the success of the welding bond, distinguishing the welded area from the unwelded area. The model was validated by comparison with the experimental data, which showed the effectiveness of the proposed FE code in predicting the cross-section morphology of the welded materials. Moreover, practical industrial information such as variation in the flyer impact velocity, collision angle, and process temperatures was predicted by varying the process laser pulse energy according to the basic principle of the process.

Designing piezoelectric stretcher through multiphysics topology for enhanced photodynamic therapy in cancer treatment

This study presents an advancement in Photodynamic Therapy (PDT) by introducing piezoelectric stretchers designed to enhance tissue surface characteristics. With a focus on mitigating the impact of irregular tissue surfaces on light distribution, we propose a structural topology optimization method to design a stretcher that utilizes the piezoelectric effect for precise tissue stretching. The aim is to achieve uniform laser irradiance across treated tissue, thereby maximizing therapy efficiency. Our approach involves the formulation of finite element models that consider the piezoelectric effect and the elastic response in a multiphysics framework. Mesh convergence tests were conducted to strike a balance between accuracy and computational efficiency in evaluating the designed piezoelectric stretcher. The investigation into the application of piezoelectric stretchers for enhancing laser irradiance on non-uniform tissue reveals significant improvements in surface uniformity with increasing electric potential. Notably, the disparity between regions of high laser focusing and defocusing decreases as the stretcher expands, achieved through the application of higher voltages. The observed enhancement from no expansion to full expansion is quantified at 31 %. The implementation of electrically driven piezoelectric tissue expansion results in a more uniform tissue surface, leading to an augmentation of laser irradiance. In the absence of expansion, distinct areas of laser focus and near-zero irradiance are noted. Conversely, an increased potential yields a predominantly even surface, with 100 % extension achieving commendable laser irradiance uniformity.

Effect of porosity and skew edges on transient response of functionally graded sandwich plates

This work for the first time presents the effect of porosity and skew edges on the transient response of functionally graded material (FGM) sandwich plates using a layerwise finite element formulation. Two configurations of FGM sandwich plates are considered. In the first configuration, the top and the bottom layers are made of the FGM and the core is made of pure metal, whereas in the second configuration, the bottom, core and the top layers are made of pure metal, FGM and pure ceramic, respectively. Four micromechanics models based on the rule of mixture are used to model porosity for these two configurations of FGM sandwich plates. A layerwise theory based on a first-order shear deformation theory for each layer that maintains the displacement continuity at the layer interface is used for the present investigation. An eight-noded isoparametric element with nine degrees of freedom per node is used to develop the finite element model (FEM). The governing equations for the present investigation are derived using Hamilton’s principle. A wide range of comparison studies are presented to establish the accuracy of the present FEM formulation. It has been shown here that the parameters like skew angle, porosity coefficient, volume fraction index, core to facesheet thickness ratio and boundary conditions have a significant effect on the transient response of FGM sandwich plates. Also, the present finite element formulation is simple and accurate.

A Finite Element Model of Unsteady Cavitating Fluid Flow around a Hydrofoil

The present study deals with the unsteady dynamics of cavitation around the NACA 0015 hydrofoil in a channel. A finite element model is proposed to solve the governing equations of momentum and mass conservation. Turbulent flows around the hydrofoil are described by the Prandtl-Kolmogorov model. The cavitation phenomenon is modeled through a mixture model involving liquid and vapor flows and the Zwart-Gerber-Belamri (ZGB) model is considered to evaluate the transport of the water vapor fraction. The variational finite element model formulation includes the mixing of the characteristic method and the finite element. Also, at the open sides of the channel flow, an open boundary condition is imposed. Numerical experiments are performed for cavitation numbers 0.8 and 0.4. The presented model predicts the essential features of unsteady cavitating flows, the generation of vapor cavities, the time-dependent oscillations of the variables and the presence of vortical flow structures associated to vapor volume concentrations during the shedding process.

A polygonal finite element formulation for modeling nearly incompressible materials

The objective of the present paper is to develop a finite element formulation for modeling nearly incompressible materials at large strains using polygonal elements. The present finite element formulation is a simplified version of the three-field mixed formulation and, in particular, it reduces the functional of the internal potential energy by expressing the field of the average volume-change in terms of the displacement field, where the latter is discretized using the Wachspress shape functions. The reduced mixed formulation eliminates the volumetric locking in nearly incompressible materials and enhances the computational efficiency as the static condensation is circumvented. A detailed implementation of the finite element formulation is presented in this study. Also, different example problems, including eigenvalue analysis, nonlinear patch test and other benchmark problems are presented for demonstrating the accuracy and the reliability of the developed formulation for polygonal elements.

A three-dimensional finite element model and image reconstruction algorithm for time-domain fluorescence imaging in highly scattering media

In this work, development and evaluation of a three-dimensional (3D) finite element model (FEM) based on the diffusion approximation of time-domain (TD) near-infrared fluorescence light transport in biological tissue is presented. This model allows both excitation and fluorescence temporal point-spread function (TPSF) data to be generated for heterogeneous scattering and absorbing media of arbitrary geometry. The TD FEM is evaluated via comparisons with analytical and Monte Carlo (MC) calculations and is shown to provide a quantitative accuracy which has less than 0.72% error in intensity and less than 37 ps error for mean time. The use of the Born–Ratio normalized data is demonstrated to reduce data mismatch between MC and FEM to less than 0.22% for intensity and less than 22 ps in mean time. An image reconstruction framework, based on a 3D FEM formulation, is outlined and simulation results based on a heterogeneous mouse model with a source of fluorescence in the pancreas is presented. It is shown that using early photons (i.e. the photons detected within the first 200 ps of the TPSF) improves the spatial resolution compared to using continuous-wave signals. It is also demonstrated, as expected, that the utilization of two time gates (early and latest photons) can improve the accuracy both in terms of spatial resolution and recovered contrast.

A 2-Dimensional Finite-Element Method for Transient Magnetic Field Computation Taking Into Account Parasitic Capacitive Effects

A formulation to model solid conductor groups in transient finite-element model (FEM) of inverter driven motor drives is presented to facilitate the users to model many solid conductors in one winding. A novel two-dimensional (2-D) FEM formulation which can include the displacement current in the direction of the model's depth is deduced. The effect of displacement current in the plane of the solution domain is represented by coupling the circuit of capacitances into the FEM equations. By introducing additional unknowns, the last set of the system equations has a symmetrical coefficient matrix. A method using electric charge as an excitation to compute the capacitance matrix is also presented to reduce the computation time. The developed algorithm has been applied to evaluate the effect of a LC filter in a converter motor drive and the parasitic capacitive effect is included.

AEROELASTIC ANALYSIS OF ROTOR BLADES USING CFD/CSD COUPLING IN HOVER MODE

A computational fluid dynamics (CFD) is coupled with a computational structural dynamics (CSD) to simulate the unsteady rotor flow with aeroelasticity effects. An unstructured upwind Navier-Stokes solver was developed for this simulation, with 2nd order time-accurate dual-time stepping method for temporal discretization and low Mach number preconditioning method. For turbulent flows, both the Spalart-Allmaras and Menter's SST model are available. Mesh deformation is achieved through a fast dynamic grid method called Delaunay graph map method for unsteady flow simulation. The rotor blades are modeled as Hodges & Dowell's nonlinear beams coupled flap-lag-torsion. The rotorcraft computational structural dynamics code employs the 15-dof beam finite element formulation for modeling. The structure code was validated by comparing the natural frequencies of a rotor model with UMARC. The flow and structure codes are coupled tightly with information exchange several times at every time step. A rotor blade model's unsteady flow field in the hover mode is simulated using the coupling method. Effect of blade elasticity with aerodynamic loads was compared with rigid blade.

Crack propagation due to time-dependent creep in quasi-brittle materials under sustained loading

This research considers static mode I fracture under sustained loading and time-dependent creep resulting from the visco-elasticity of the bulk material outside the fracture process zone, and cohesive crack rate dependence. Under sustained loading, the magnitude of the bulk creep deformations are taken to be related to the elastic deformations, material parameters and time. The finite element model formulation used for the analysis of quasi-brittle fracture of Attard and Tin-Loi [M.M. Attard, F. Tin-Loi, Numerical simulation of quasibrittle fracture in concrete, Engrg. Fract. Mech. 72 (2005) 387–411] is extended to include time-dependent behaviour. Fracture is lumped and assumed to occur only at interface nodes by means of a softening-fracture law. The formulation is based on an assembled structural piecewise-linear constitutive law and is set in the form of a “linear complementarity problem” (LCP). In order to include time effects, two main sources of time dependence including bulk creep and crack rate dependency are taken into consideration. The simple concept of a creep coefficient is adopted to account for bulk creep. A simplified crack rate model is proposed and used to represent rate-dependent crack opening. The model is applied to predicting the time-dependent crack propagation in quasi-brittle materials subjected to sustained loading. The experimental results of Zhou [F.P. Zhou, Time-dependent crack growth and fracture in concrete, Doctoral Dissertation, Lund University, Lund, Sweden, 1992] are used to compare with the results of simulations. The crucial aspects of time-dependent behaviour are simulated and presented.

Three-dimensional morphometry of the femoral condyles

Objective. The present study describes the geometry of the three-dimensional articular surfaces of the human femoral condyles based on measurements of surface coordinates. Design. The purpose was not to obtain a complex representation of one single condyle, but to describe the femoral condyles using simple geometric parameters based on measurements using a number of specimens. Background. In joint modeling, a representative knee joint geometry is often desired which requires an approximation of the irregular joint geometry while taking into account interspecimen variations. Methods. An optical device was used to measure the condylar articular surfaces of 12 human femurs in the femorotibial contact region. The sagittal profiles were reconstructed by means of two circular arcs and the radial profiles by means of one circular arc. Results. The results provide the geometric parameters necessary for the three-dimensional reconstruction of the articular surfaces of the femoral condyles. The results indicate that the medial and lateral condyles of the distal femur are significantly asymmetric in a number of morphological features. Conclusion. The primary application of the results is expected to be in the formulation of finite element models of the knee joint for static contact problems. Relevance Numerical models of the knee joint are being widely used to study the mechanics of the joint. However, formulation of such models demands a prior knowledge of the complex three-dimensional geometry of the articular surfaces of the natural joint to establish the input parameters of the model.

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part I: Flexural–torsional stability

AbstractIn formulating a finite element model for the flexural–torsional stability and 3‐D non‐linear analyses of thin‐walled beams, a rotation matrix is usually used to obtain the non‐linear strain–displacement relationships. Because of the coupling between displacements, twist rotations and their derivatives, the components of the rotation matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the rotation matrix. A simplified small rotation matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin‐walled beams of open cross‐section. However, the approximations in the small rotation matrix may lead to the loss of some significant terms in the stability stiffness matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin‐walled beams, while a companion paper investigates the effects of approximations in the 3‐D non‐linear analysis. It is found that a finite element line model based on a small rotation matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin‐walled beams, modification of the model is needed, or a finite element model based on a second‐order rotation matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part I: Flexural–torsional stability

In formulating a finite element model for the flexural–torsional stability and 3-D non-linear analyses of thin-walled beams, a rotation matrix is usually used to obtain the non-linear strain–displacement relationships. Because of the coupling between displacements, twist rotations and their derivatives, the components of the rotation matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the rotation matrix. A simplified small rotation matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin-walled beams of open cross-section. However, the approximations in the small rotation matrix may lead to the loss of some significant terms in the stability stiffness matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin-walled beams, while a companion paper investigates the effects of approximations in the 3-D non-linear analysis. It is found that a finite element line model based on a small rotation matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin-walled beams, modification of the model is needed, or a finite element model based on a second-order rotation matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.

Finite element model tuning using automated structural optimization system software

A method of adjusting analytical finite element models to measured data is presented. The algorithm uses a mathematical optimization strategy to minimize deviations between measured and analytical modal frequencies and partial mode shapes. A mode tracking algorithm is used to identify and account for mode switching during the optimization process. The algorithm was successfully implemented using the Automated Structural Optimization System Software. Experimental results are presented for tuning a lightly damped 6-m flexible frame structure. The results demonstrate excellent agreement between the tuned model and measured data and illustrate the importance of off-nominal validation before accepting a model for simulation purposes. CCURATE prediction and simulation of the dynamic behav- ior of large flexible space structures require analytical models that agree with measured data. Unfortunately, uncertainty in a finite element model (FEM) implies less than perfect correlation between analytical and measured data. When the disagreement is deemed unacceptable, it is necessary for the design engineer to make ad- justments to the FEM. For large problems the number of potential parameters to adjust, such as elemental areas, elastic moduli, inertia moments, etc., quickly becomes overwhelming. Thus a systematic method is required to ensure the adjustments produce the desired results. To this end, a method is introduced that poses a numerical optimization problem, namely, given a set of measured eigenvalues and partial eigenvectors, determine the values of selected physical parameters of the model that minimize the weighted deviations from the analytical eigenvalues and eigenvectors. Numerous techniques and goals of model tuning, also referred to as model refinement or model identification, have appeared in the literature. The common attribute of these techniques is that they at- tempt to minimize the required modification to the analytical mass and stiffness matrices, assuming the FEM is a reasonable approxi- mation to the physical structure. Sensitivity based approaches have been presented in the literature that adjust the matrices by estab- lishing an objective function based on the difference between the experimental and analytical model data.15 An advantage of those methods is that the updated models are consistent with the FEM formulation, and thus the connectivity is preserved. An alternative method employing an optimal matrix update has been developed by Berman and Nogy6 and Baruch.7 In this method, a perturbation mass, damping, or stiffness matrix is determined that, when added to the analytical matrices, produces the measured result. The advantage of this method is a tuned analytic model that exactly reproduces the experimental data. Its shortcoming is that it does not guarantee the closeness to unmeasured modes not used in the tuning process. This is a result of potentially unrealistic changes in the stiffness matrix, such as the introduction of load paths that physically do not exist. To overcome some of the shortcomings of Baruch's method of stiff- ness matrix adjustments, Kabe8 and Kammer9 introduced objective functions that ensure stiffness terms are corrected in a manner such that the connectivity of the analytical model is preserved. Similar

Un modelo numérico utilizando el método de los elementos finitos para la simulación de flujo monofásico en dos dimensiones

A Finite Element Model (FEM) is presented to simulate flow of a single phase fluid through a porous media in 2-dimensions. This model is the initial phase of a research proiect that builds on the development of a general reservoir simulator for three phase flow. The numerical model can be applied lo heterogeneous fields with constant thickness but complex irregular geometry. No restrictions are imposed in the number of wells in the field. Governing equations and the boundary conditions are specified, followed by a description of the FEM formulation. Analytical solutions are used to verify the predictions of the model in a circular reservoir. lt is also applied in the preliminary simulation of a sector in the Orito oil field (Ecopetrol) located in the Putumayo_Department (Southern Colombia).

Three-Dimensional Modeling of Hydraulic Fractures in Layered Media: Part II—Calibrations, Parametric Sensitivity and Field Simulations

A comprehensive finite element model formulation for the simulation of vertical hydraulic fracture propagation in a multi-layered reservoir has been presented in Part I of the companion paper (Advani et al., 1990). This paper presents a numerical calibration of this model by comparing results with selected benchmark cases. Parametric sensitivity investigations are also performed to illustrate the role of different variables on hydraulic fracture configuration evolution in multi-layered formations. In addition, field simulations for a multi-layered reservoir with differential in-situ stress and mechanical property contrasts are conducted to illustrate the scope of the developed model.

A direct formulation of finite element models in mechanics

This paper offers a simple valid alternative approach to the formulation of finite element models. Using two‐dimensional steady state heat conduction as a representative problem, the paper formalizes and extends a direct approach, normally restricted to constant strain triangular (CST) elements, to the construction of general finite element models. The approach is illustrated with the conventional 4, 8, and 9 noded quadrilateral elements. The sufficient conditions for convergence are explained in straightforward physical terms. The direct approach is based on the dual roles of shape functions, and does not require the use of variational methods or functional analysis. The formulation is shown to be equivalent to a weak form of Galerkin which makes explicit use of three forms of residual. This approach is intended to enable engineering students to gain a sound understanding of basic finite element methods with a minimum of mathematics.

Finite element analysis of magnetic fields with permanent magnet

This paper describes a finite element formulation for modeling permanent magnets in electromagnetic devices. The method was used to predict flux densities of air stablized electrical machine under no load condition. Search coil measurements of flux densities in the device were used to verify the validity of the model, and an agreement well within 5% was obtained.

On the accuracy of finite element solutions of problems with traction boundary conditions

In the variational formulation of finite element models for mechanics problems, boundary conditions are classified as either essential or natural. The traction boundary conditions are identified as natural in many commonly-used finite element models. A valid finite element mode must satisfy explicitly the essential boundary conditions, since the latter are satisfied implicitly as a posteriori conditions of the formulation. Though assumed solutions can be arbitrary regarding natural boundary conditions, the arbitrariness can affect the accuracy of solutions, because the weighted error residuals of all the Euler equations of the variational principle should be minimized to obtain solutions. As a possible alternative to refining finite element meshes for improving the accuracy of solutions, an explicit enforcement of the natural boundary conditions can be considered. In this work, the nature of the implicit enforcement of the natural boundary conditions of a problem through a variational formulation is investigated. Systematic methods to enforce natural boundary conditions explicitly to improve the accuracy of solutions are developed for two finite element models. Consider a boundary value problem defined by VZu = f in a domain D and some boundary conditions at boundary B. The problem has a variational principle that the stationary condition of a scalar functional I with respect to an admissible solution u (which satisfies u = h on B, if such a condition exists) yields the interior equation and a natural boundary condition, au/an = g on B where n is the outward normal to B. The functional I is:

Solution procedures for nonlinear structural dynamic analysis

Abstract A review of the solution techniques most widely used in nonlinear structural dynamics is presented. For nonlinear transient responses several explicit and implicit direct integration methods are compared with respect to accuracy, stability and computational efficiency. It is concluded that the choice of a suitable method depends upon the nature of the method, the formulation of finite element models, and the problem itself. In general, the Park method seems to be superior to the others in nonlinear dynamic analysis. If equilibrium iterations are performed at each time step, the Newmark — s= 1/4 method should be preferred.

Finite elements on the basis of continuum mechanics

AbstractThe formulation of finite element models on the basis of different variational principles is reviewed. The degrees‐of‐freedom of the elements are defined in an abstract way without the help of nodal points. In this manner it is possible to describe elements of arbitrary shape and accuracy. The formulation is confined to linear elasto‐statics. For two‐dimensional structures two hybrid element models are developed using Legendre polynomials on the element boundaries. Examples of plane stress problems are used to test the generation of convergence by increasing the accuracy of the elements vs. by increasing the number of elements.