Steady-state Finite Element Method Research Articles

Application of the B-Spline Method to Solve Nonlinear Problem of Heat Conduction With Radiation Boundary Conditions Using Kirchhoff Transformation

Abstract This paper concerns the joint application of the B-spline method and the Kirchhoff transformation to solve the nonlinear problem of thermal conduction with radiation type boundary conditions. The proposed method requires few iterations, sometimes none, for solids subjected to prescribed temperature boundary conditions. This method can be deployed by other numerical approaches (boundary element method, finite element method, finite element method, etc.) for the resolution of the heat conduction equation (linear or nonlinear), in terms of the Kirchhoff transformation θ. For numerical implementation, the steady-state finite element method is considered. The numerical validation was performed for a hollow aluminum cylinder whose outer surface is subjected to radiation. Three types of thermal conductivities are considered: (i) constant, (ii) linear, and (iii) nonlinear. As an application, we studied the thermal response of an aluminum reactor, in the form of an annular disk with cooling tubes, exposed to thermal radiation.

Research Paper: Numerical Simulation of Thermo-elastic Stresses in Germanium Crystal Grown by Czochralski Technique during Different Lengths of Crystal

In the process of crystal growth by Czochralski technique, lower part and core of the crystal are warmer than other parts of crystal and its environment, which leads to expansion in different parts of the crystal. The result of this thermal gradient is strain, which eventually causes thermo-elastic stress in the crystal. Increasing this stress leads to transition of the material from elastic limit and entering plastic area. To show thermo-elastic stress in crystals, a criterion called Von Misses stress is used. Using the solid mechanics approach, the mechanical response of crystal to the stresses can be determined through appropriate structural equations. In this paper, using appropriate structural equations, a set of numerical simulations of temperature field, thermal stress and dislocation density for a Czochralski setup used to grow Ge single crystal have been done for different heights of crystal. In order to investigate dislocation density, using a simple first-order approximation, in which the dislocation density is proportional to radial gradient of temperature is used. A two-dimensional steady state finite element method has been applied for all calculations. The numerical results reveal that the thermal field and thermal stress are mainly dependent on the crystal height, heat radiation and gas flow in the growth system. As the height of the crystal increases and the shape of the crystal-melt interface changes, we see an increase in thermo-elastic stress and dislocation density.

Effect of the ceramic tube shape on global heat transfer, thermal stress and crystallization front in low thermal gradient (LTG) Czochralski growth of scintillating BGO crystal

The effect of ceramic tube on thermal field, thermal stress and crystallization front have been experimentally and numerically analyzed for a low thermal gradient (LTG) Czochralski growth of crystal. The governing equations have been solved using 2D steady state finite element method. To do it, a real growth setup with three different shapes of the ceramic tube (cylindrical, legged cylindrical and conical) were simulated in order to compare the numerical results with experimental findings of crystal–melt interface. It was found that the best LTG conditions above the melt surface, namely, the lowest thermal stress and the crystallization front with the smallest deflection height, relate to the case of conical ceramic tube due to the lowest internal radiation exchange within the BGO crystal.

Response optimization of four stroke variable compression ratio diesel engine cylinder head with stress analysis

The main aim of this work is to predict the design performance based on the stress/strain and behavior of cylinder head under various operating conditions. The effects of engine operating conditions such as combustion gas maximum internal pressure and temperature on the cylinder head have been analyzed. The maximum gas pressure vs crank angle are taken from the experimental set up of a variable compression ratio (VCR) Diesel engine . The analysis was carried out at three different compression ratios like 16.5,17.5,18.5 using a finite element analysis (FEA) software package, which is use to simulate and predict the Von-Mises stresses and strain pattern and thermal distribution of the cylinder head structure .The geometrical modelling was carried out using CATIA and analysis is done on ANSYS software. In this investigation, structural and thermal analysis of the cylinder head highlight several areas of interest. The maximum stress is found not exceeding the material strength of cylinder head, and thus the basic design criteria, namely no yielding and no structural failure under firing load case are satisfied. This steady-state finite element method (FEM) stress analysis can play a very effective role in the rapid prototyping of the cylinder head. The results of the thermo-mechanical analysis show that when the engine is running, the stress in the region is compressive, caused by thermal loading and combustion pressure. The fatigue analysis is carried out to know the factor of safety and life of the cylinder head it indicates maximum life of 1x106 cycles at all compression ratios are determined through finite element analysis. Further response optimization method is employed for the cylinder head by varying the thickness of the head in order to minimize the various stresses and deformation and temperature distribution values

Computational analysis of heat transfer, thermal stress and dislocation density during resistively Czochralski growth of germanium single crystal

In this paper, a set of numerical simulations of fluid flow, temperature gradient, thermal stress and dislocation density for a Czochralski setup used to grow IR optical-grade Ge single crystal have been done for different stages of the growth process. A two-dimensional steady state finite element method has been applied for all calculations. The obtained numerical results reveal that the thermal field, thermal stress and dislocation structure are mainly dependent on the crystal height, heat radiation and gas flow in the growth system.

State of the Art in Rolling Process Modelling

Pushed forward by mill complexification and harder product quality requirements, the mathematical treatment has evolved into very complex, multi-coupled models: roll deformation, thermal transfer, lubrication, and oxide scales are a few examples. The diversity of rolled products and rolling mills makes rolling process modelling a vast field indeed. Beyond general, necessarily very costly models, partial descriptions are developed for more targeted goals. Three methods share the market: the Slab Method (SM), the Upper Bound Method (UBM), and the Finite Element Method (FEM). A few modelling strategies based on the above three are described, providing accurate solutions in an industry-compatible computation time: steady-state FEM modelling; faster 2D SM with large roll flattening to build thin sheet or temper rolling force models; low-cost 3D roll stack deformation package for profile/flatness of sheets; FEM results-based UBM models of width variations in the Tandem Cold Mill; post-bite profile and flatness evolution (interstand behavior); FEM description of oxide scale behavior in descaling; SM with modern lubrication modelling. Evolutions of rolling processes are questioned to point to new demands on modelling and how to answer them.

Fully Coupled Dynamic Model of Thomson’s Levitating Ring

The complete electromagnetic-mechanical-thermal modeling approach is demonstrated on the Thomson’s levitating ring experiment. This problem, but with a different set of parameters, was also described by Laithwaite and defined by Freeman and Lowther as TEAM17 problem. This paper presents its most complete solution. The factors for determining the vertical displacements and influence of the height of the ring are explained. The whole apparatus is explained in detail so that an accurate computer model can be made. The model introduced is validated against measurements and transient finite- element method (FEM) solvers. All the results prove the applicability of using steady-state FEM analysis with lumped parameter approach for a fast and effective fully coupled modeling.

Characteristics of an Electrodynamic Wheel Using a 2-D Steady-State Model

The mechanical rotation of a radially positioned permanent-magnet Halbach array above a conducting, nonmagnetic track induces eddy currents in the track that can inductively create suspension and propulsion forces simultaneously. The parameters that affect the performance of this electrodynamic wheel are studied using a 2-D steady-state finite-element method. Tradeoffs between the lift and thrust force performance are investigated and methods to improve the thrust efficiency are proposed.

Characteristics of CF/PEI tape winding process with on-line consolidation

The effects of processing conditions on consolidation quality for the tape winding process of thermoplastic composites with on-line consolidation have been investigated. Composite rings were manufactured using carbon fibre (CF)/polyetherimide (PEI) tape for both cases with and without an insulated ring. Based on heat transfer and intimate contact/autohesion mechanisms, a steady-state finite element method (FEM) model was developed to analyze consolidation quality and overheating in composite tapes. The processing windows with the upper bound for good consolidation and the lower bound for overheating were established for winding speed as a function of processing temperature, when the compaction pressure was kept constant. Good correlation was obtained between the steady-state FEM model and experimental data.

Development of a FEM Reservoir Model Equipped with an Effective Permeability Tensor and its Application to Naturally Fractured Reservoirs

The numerical simulation of naturally fractured reservoirs is commonly based on the continuum dual-porosity concept. However, the model needs to assume that the fractures form very regular patterns, different from field observations. In order to overcome this problem, we proposed a method that can consider the characteristics of the real fracture system and developed a two-phase transient finite element method (FEM) (TENFEM) model able to implement an effective permeability tensor. The permeability tensors were estimated by using a single-phase, steady-state FEM (EPC) model also coded in this work, where Darcy's law in matrix and cubic law in fracture were adopted for considering the flow characteristics in the discrete fracture network (DFN) approach. The developed models were applied to the paludal sand reservoir of a multiwell experiment (MWX) site to demonstrate validity and applicability of the models. The estimated average permeability in the MWX is almost identical compared to the results of the well test analysis. Then the numerical simulation with the TENFEM model was performed by using the transient pressure recorded from a gas well test on the MWX #1. From the results of bottom hole pressure and pressure distribution within the system, it was noted that the model with effective permeabilities generated almost identical results against the DFN model in the aspects of the behavior and the direction of fluid propagation with pressure decline, and therefore the proposed model is considered to be more efficient in terms of the computation time as well as the required storage capacity. Also from the upscaling study, it was shown that even with fewer numbers of grid blocks the developed model could effectively describe the directional flow caused by the fracture characteristics such as orientation and density of the fracture.

A three-dimensional steady-state finite element analysis of square die extrusion by using automatic mesh generation

Steady-state finite element analysis is made for three-dimensional hot extrusion of sections through square dies by using an automatic mesh generator which can generate three-dimensional meshes by shifting two-dimensional meshes. In industrial practice, the design of extrusion dies is still an art rather than science, especially for complicated profiles, because the die design for a new extrusion is developed from previous experience and in-plant trials. The objective of this study is to develop a steady-state finite element method for hot extrusion through square dies, and to provide a theoretical basis for an optimal die design and process control for the extrusion technology. In the present investigation, steady-state assumption is used for both the analyses of deformation and temperature. The analysis of temperature distribution includes heat transfer, and is carried out by decoupling from the analysis of deformation. Convection link element is adopted for the heat transfer analysis between the billet and the container, and also between the billet and the die. Computations are carried out for solid and hollow extrusion of several sections. The present method of analysis has been shown to provide good results comparable with the non-steady-state method with reduced computation time. Distributions of temperature, effective strain rate, velocity and mean stress are discussed for effective design of an extrusion die.