Introduction to Finite Elements in Engineering

2021 ◽  
Author(s):  
Tirupathi Chandrupatla ◽  
Ashok Belegundu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Time Dependent ◽  
Plate Bending ◽  
The Finite Element Method ◽  
Hands On ◽  
Sparse Solutions ◽  
2D And 3D ◽  
Time Dependent Problems ◽  
Element Method

Thoroughly updated with improved pedagogy, the fifth edition of this classic textbook continues to provide students with a clear and comprehensive introduction the fundamentals of the finite element method. New features include coverage of core topics – including mechanics and heat conduction, energy and Galerkin approaches, convergence and adaptivity, time-dependent problems, and computer implementation – in the context of simple 1D problems, before advancing to 2D and 3D problems; expanded coverage of reduction of bandwidth, profile and fill-in for sparse solutions, time-dependent problems, plate bending, and nonlinearity; over thirty additional solved problems; and downloadable Matlab, Python, C, Javascript, Fortran and Excel VBA code providing students with hands-on experience. Accompanied by online solutions for instructors, this is the definitive text for senior undergraduate and graduate students studying a first course in the finite element method, and for professional engineers keen to shore up their understanding of finite element fundamentals.

A New Mixed Finite Element–Differential Quadrature Formulation for Forced Vibration of Beams Carrying Moving Loads

2010 ◽  
Vol 78 (1) ◽  
Author(s):  
A. A. Jafari ◽  
S. A. Eftekhari
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Gaussian Elimination ◽  
Mixed Finite Element ◽  
Differential Quadrature ◽  
Time Dependent ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Time Dependent Problems ◽  
Element Method

In this paper, a new version of mixed finite element–differential quadrature formulation is presented for solving time-dependent problems. The governing partial differential equation of motion of the structure is first reduced to a set of ordinary differential equations (ODEs) in time using the finite element method. The resulting system of ODEs is then satisfied at any discrete time point apart and changed to a set of algebraic equations by the application of differential quadrature method (DQM) for time derivative discretization. The resulting set of algebraic equations can be solved by either direct methods (such as the Gaussian elimination method) or iterative methods (such as the Gauss–Seidel method). The mixed formulation enjoys the strong geometry flexibility of the finite element method and the high accuracy, low computational efforts, and efficiency of the DQM. The application of the formulation is then shown by solving some moving load class of problems (i.e., moving force, moving mass, and moving oscillator problems). The stability property and computational efficiency of the scheme are also discussed in detail. Numerical results show that the proposed mixed methodology can be used as an efficient tool for handling the time-dependent problems.

scholarly journals A Map-plane Finite-element Model: Three Modeling Experiments

1989 ◽  
Vol 35 (119) ◽  
pp. 48-52 ◽  
Author(s):  
James L. Fastook ◽  
Judith E.. Chapman
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Ice Sheet ◽  
Time Dependent ◽  
Element Formulation ◽  
Initial Surface ◽  
The Finite Element Method ◽  
Sheet Flow ◽  
Flow Law ◽  
Element Method

AbstractPreliminary results are presented on a solution of the two-dimensional time-dependent continuity equation for mass conservation governing ice-sheet dynamics. The equation is solved using a column-averaged velocity to define the horizontal flux in a finite-element formulation. This yields a map-plane model where flow directions, velocities, and surface elevations are defined by bedrock topography, the accumulation/ablation pattern, and in the time-dependent case by the initial surface configuration. This alleviates the flow-band model limitation that the direction of flow be defined and fixed over the course of the modeling experiment. The ability of the finite-element method to accept elements of different dimensions allows detail to be finely modeled in regions of steep gradients, such as ice streams, while relatively uniform areas, such as areas of sheet flow, can be economically accommodated with much larger elements. Other advantages of the finite-element method include the ability to modify the sliding and/or flow-law relationships without materially affecting the method of solution.Modeling experiments described include a steady-state reconstruction showing flow around a three-dimensional obstacle, as well as a time-dependent simulation demonstrating the response of an ice sheet to a localized decoupling of the bed. The latter experiment simulates the initiation and development of an ice stream in a region originally dominated by sheet flow. Finally, a simulation of the effects of a changing mass-balance pattern, such as might be anticipated from the expected carbon dioxide warming, is described. Potential applications for such a model are also discussed.SYMBOLS USEDa(x,y) Accumulation/ablation rate.A Flow-law parameter.B Sliding-law parameter.CijC Global capacitance matrix.f Fraction of the bed melted.Fij,F Global load vector.g Acceleration of gravity.hj,h Ice-surface elevation.H Ice thickness.k(x,y) Constitutive equation constant of proportionality.kij Global stiffness matrix.m Sliding-law exponent.n Flow-law exponent.ρ Density of ice.σ(x,y) Ice flux.t Time.U Column-average ice velocity.UF Column-average deformation (flow) velocity.US Sliding velocity.v Variational trial function.x,y Map-plane coordinates.

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