TAILIEUCHUNG - Ebook Finite element method (second edition): Part 2

(BQ) Part 2 book "Finite element method" has contents: Higher-order elements - The isoparametric concept, further topics in the finite element method; convergence of the finite element method; the boundary element method; computational aspects. | 4 Higher-order elements: the isoparametric concept A two-point boundary-value problem In Chapter 3, the finite element philosophy was developed and illustrated by means of examples in which the interpolation functions were linear polynomials. There is no reason to choose linear functions only; indeed, higher-order polynomials are frequently used and in principle cause no more difficulty. We shall illustrate the approach using the Galerkin finite element method of Sections –. Example −u = f (x), 0 < x < 1, u(0) = 1, u (1) + 2u(1) = 3. Suppose that the region 0 ≤ x ≤ 1 is divided into E elements, each element having three nodes; a typical element is shown in Fig. . In terms of its three nodal values, the variable u may be uniquely interpolated as a quadratic polynomial, the interpolation being given by ue = Ne Ue , T where Ue = [U1 U2 U3 ] and the shape function matrix is Ne = 1 2 ξ2 − ξ 2(1 − ξ 2 ) ξ2 + ξ . The elements of Ne are the usual quadratic Lagrange interpolation polynomials; see Fig. . The development of the method follows in exactly the same way as in Section , and we use the notation of Section to obtain the relevant element matrices. 1 2 xm h 3 x = 2 (x – xm ) h Fig. The three-node element for the problem of Example . 142 The Finite Element Method Ne (ξ) i 1 Ne (ξ) 2 Ne (ξ) 1 Ne (ξ) 3 0 − −1 − − − − 0 1 ξ Fig. Lagrange quadratic interpolation polynomials. Thus, in this case, αe = [d/dx] Ne = 1 [2ξ − 1 h − 4ξ 2ξ + 1]. Therefore () 1 ke = 2 h ⎤ 1 − 4ξ + 4ξ 2 4ξ − 8ξ 2 −1 + 4ξ 2 h ⎣ 16ξ 2 −4ξ − 8ξ 2 ⎦ dξ, 2 −1 sym 1 + 4ξ − 8ξ 2 1 ⎡ . 1 2 3 ⎤ 7 −8 1 1 1 ⎣ ke = 16 −8 ⎦ 2 . 3h sym 7 3 ⎡ The element force vector is given by ¯e = f 1 f Ne −1 fh = [1 6 when f (x) ≡ f , a constant. T h dξ 2 T 4 1] 143 Higher-order elements: the isoparametric concept For boundary elements, there are also contributions to the stiffness .

• Cơ khí - Chế tạo máy
• Finite element method
• The isoparametric concept
• Higher order elements
• The finite element method
• The boundary element method
• Computational aspects
• Further topics
• Lecture Finite element method
• Basis equation of plane problem
• Finite element discretization
• Quadrilateral plane element
• Triangular plane element
• Introduction to the finite element method
• Finite element equations
• Heat transfer
• Fem for solid mechanics problems
• Finite elements
• The finite element method in engineering
• Interpolation models
• Isoparametric elements
• System equations
• Introduction to mechanics
• Fundamentals for finite element method
• FEM for beams
• FEM for frames
• FEM for Two Dimensional solids
• Finite element method using MATLAB
• CRC Mechanical engineering series
• Approximation techniques
• Finite element programming
• Direct approach
• Vietnam Journal of Mechanics
• Smoothed finite element method
• Two dimensional elasto plasticity
• The finite element
• The smoothed finite element method
• Mesh distortion and volumetric locking
• Adaptive finite element refinement
• The time dimension
• Couple systems
• Finite element analysis
• Computer procedures
• Locking phenomena in finite element analysis
• Deep beam and removal method
• Exhibit slow convergence
• Finite Element Methods
• Bài giảng Phương pháp phần tử hữu hạn
• Stiffness method
• Stiffness matrix
• Boundary conditions
• Lecture Structural analysis
• Finite element
• Structural analysis
• Finite element approach
• Element approach
• A first course in the finite element method
• Introduction to the stiffness method
• Development of truss equations
• Development of beam equations
• The plane stress
• Numerical method
• Electro Magnetic analysis
• Thermal analysis
• Axisymmetric elements
• Isoparametric formulation
• Three dimensional stress analysis
• Plate bending element
• Vibrational principles
• Galerkin approximation
• Partial differential equations
• Isoparametric finite elements
• Historical introduction
• Weighted residual
• Variational methods
• Elliptic problems
• Heat and fluid flow
• Some basic discrete systems
• Steady state heat conduction
• Transient heat conduction analysis
• Truss structures
• Elasticity problem
• Basic equations
• Solution procedure
• Three dimensional problems
• Dynamic analysis
• Cast3m implementation of the extended finite element method
• The extended finite element method for cohesive crack
• Quasi brittle materia
• The proposed implementation
• The displacement discontinuities
• The node based smoothed finite element method
• Free vibration problems
• The proposed formulation
• No spurious non zero energy modes
• Science and technology development
• Extended finite element method
• Simulating the mechanical behavior
• Multiple random holes and inclusions
• Functionally graded material
• The finite element method software
• Tạp chí khoa học và công nghệ
• Modeling the mechanical behavior
• Functionally graded material plates with multiple random inclusions
• Modeling of electromagnetic force
• Magnetic vector potential formulation via
• Sbproblem finite element method
• Subproblem finite element method
• Magnetic vector potential formulation