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The Galerkin Method

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The Galerkin Method Institute of Structural Engineering Page 1 Chapter 3 Variational Formulation & the Galerkin Method Method of Finite Elements I Institute of Structural Engineering Page 2 Today’s Lecture Contents: • Introduction • Differential formulation • Principle of Virtual Work • Variational formulations • Approximative methods • The Galerkin Approach Method of Finite Elements I Institute of Structural Engineering Page 3 The differential form of physical processes Physical processes are governed by laws (equations) most probably expressed in a differential form: • The axially loaded bar equation: • The isotropic slab equation: • The Laplace equation in two dimensions: (e.g. the heat conduction problem) Method of Finite Elements I Institute of Structural Engineering Page 4 The differential form of physical processes Focus: The axially loaded bar example. Consider a bar loaded with constant end load R. Strength of Materials Approach Equilibrium equation R R fx()= R⇒σ ( x) = A x Constitutive equation (Hooke’s Law) σ (x) R ε ()x == R E AE f(x) Kinematics equation L-x δ (x) ε x = ( ) x Given: Length L, Section Area A, Young's modulus E Find: stresses and deformations. Rx Assumptions: δ x = The cross-section of the bar does not change after loading. ( ) AE The material is linear elastic, isotropic, and homogeneous. The load is centric. End-effects are not of interest to us. Method of Finite Elements I Institute of Structural Engineering Page 5 The differential form of physical processes Focus: The axially loaded bar example. The Differential Approach Consider and infinitesimal element of the bar: Equilibrium equation Δσ dσ Aσ = A(σ + Δσ ) ⇒ A lim = 0 ⇒ A = 0 ! x dx Δx→0 Δ Constitutive equation (Hooke’s Law) σε= E Kinematics equation du ε = dx d 2u AE = 0 Strong Form dx2 Boundary Conditions (BC) Given: Length L, Section Area A, Young's modulus E Find: stresses and deformations. u(0) = 0 Essential BC Assumptions: σ (L) = 0 ⇒ The cross-section of the bar does not change after loading. du The material is linear elastic, isotropic, and homogeneous. AE = R Natural BC The load is centric. dx x=L End-effects are not of interest to us. Method of Finite Elements I Institute of Structural Engineering Page 6 The differential form of physical processes Focus: The axially loaded bar example. The Differential Approach d 2u AE = 0 Strong Form dx2 R u(0) = 0 Essential BC du x AE = R Natural BC dx x=L Definition The strong form of a physical process is the well posed set of the underlying differential equation with the accompanying boundary conditions Method of Finite Elements I Institute of Structural Engineering Page 7 The differential form of physical processes Focus: The axially loaded bar example. The Differential Approach d 2u AE = 0 Strong Form dx2 R u(0) = 0 Essential BC du x AE = R Natural BC dx x=L v This is a homogeneous 2nd order ODE with known solution: Analytical Solution: du(x) u(x) = uh = C1x + C2 & ε(x) = = C1 = const! dx Method of Finite Elements I Institute of Structural Engineering Page 8 The differential form of physical processes Focus: The axially loaded bar example. The Differential Approach d 2u AE = 0 Strong Form dx2 R u(0) = 0 Essential BC du x AE = R Natural BC dx x=L Analytical Solution: To fully define the solution (i.e., to evaluate the values of parameters C 1 , C 2 ) we have to use the given boundary conditions (BC): C2 = 0 u(0)=0 u(x) = uh = C1x + C2 ⎯d⎯u ⎯R⎯→ R = dx EA C = x=L 1 EA Rx ⇒ u(x) = v Same as in the mechanical approach! AE Method of Finite Elements I Institute of Structural Engineering Page 9 The Strong form – 2D case A generic expression of the two-dimensional strong form is: and a generic expression of the accompanying set of boundary conditions: : Essential or Dirichlet BCs : Natural or von Neumann BCs Disadvantages The analytical solution in such equations is i. In many cases difficult to be evaluated ii. In most cases CANNOT be evaluated at all. Why? • Complex geometries • Complex loading and boundary conditions Method of Finite Elements I Institute of Structural Engineering Page 10 Deriving the Strong form – 3D case su su : supported area with prescribed displacements U s f s f : surface with prescribed forces f f B : body forces (per unit volume) U : displacement vector ε : strain tensor (vector) σ : stress tensor (vector) Method of Finite Elements I Institute of Structural Engineering Page 11 Deriving the Strong form – 3D case Kinematic Relations LU ε = ⎡ ∂ ⎤ ⎢ 0 0 ⎥ ⎢ ∂x ⎥ ⎢ ∂ ⎥ 0 0 ⎢ y ⎥ ⎢ ∂ ⎥ ⎢ ∂ ⎥ ⎡ U ⎤ ⎢ 0 0 ⎥ ⎢ ⎥ ∂z T ⎡ ⎤ U = V , L = ⎢ ⎥, ε = ε xx ε yy ε zz 2ε xy 2ε yz 2ε xz ⎢ ⎥ ⎢ ∂ ∂ ⎥ ⎣⎢ ⎦⎥ ⎢ W ⎥ 0 ⎣ ⎦ ⎢ ∂y ∂x ⎥ ⎢ ⎥ ⎢ ∂ ∂ ⎥ 0 ⎢ ∂z ∂y ⎥ ⎢ ⎥ ⎢ ∂ ∂ ⎥ 0 ⎢ ∂z ∂x ⎥ ⎣ ⎦ Method of Finite Elements I Institute of Structural Engineering Page 12 Deriving the Strong form – 3D case Strain Compatibility – Saint Venant principle ⎡⎤∂∂22 ∂ 2 02− 0 0 ⎢⎥yx22 xy ⎢⎥∂∂ ∂∂ ⎢⎥∂∂22 ∂ 2 ⎢⎥002022 − ⎢⎥∂∂zy ∂∂ yz ⎢⎥∂∂22 ∂ 2 ⎢⎥0002− ∂∂zx22 ∂∂ xz L = ⎢⎥ L1ε = 0 ⇒ 1 ⎢⎥∂∂∂∂2222 ⎢⎥00−−2 ⎢⎥∂∂yz ∂∂ xz ∂ x ∂∂ xy ⎢⎥∂∂∂∂2222 ⎢⎥00−− ⎢⎥∂∂xz ∂∂ zy ∂∂ xy ∂ y2 ⎢⎥22 2 2 ⎢⎥∂∂ ∂ ∂ 00 2 −− ⎣⎦⎢⎥∂∂xy ∂ z ∂∂ xz ∂∂ yz Method of Finite Elements I Institute of Structural Engineering Page 13 Deriving the Strong form – 3D case Equilibrium Equations B Body loads: L2τ+f =0 τTT, LL= = ⎣⎦⎡⎤ττττττxx yy zz xy yz zx 2 Surface loads: ⎡⎤lmn00 0 s on smln we have Ντ− ff = 0 where N =⎢⎥0 0 0 f ⎢⎥ ⎣⎦⎢⎥00nml 0 l, m and n are cosines of the angles between the normal on the surface and X, Y and Z Method of Finite Elements I Institute of Structural Engineering Page 14 Deriving the Strong form – 3D case Constitutive Law τ= Cε C is elasticity matrix and depends on material properties E and ν (modulus of elasticity and Poisson’s ratio) ε= C−1τ Method of Finite Elements I Institute of Structural Engineering Page 15 Differential Formulation Summary - General Form 3D case: • Boundary problem of the linear theory of elasticity: differential equations and boundary conditions • 15 unknowns: 6 stress components, 6 strain components and 3 displacement components • 3 equilibrium equations, 6 relationships between displacements and strains, material law (6 equations) • Together with boundary conditions, the state of stress and deformation is completely defined Method of Finite Elements I Institute of Structural Engineering Page 16 Principle of Virtual Work • The principle of virtual displacements: the virtual work of a system of equilibrium forces vanishes on compatible virtual displacements; the virtual displacements are taken in the form of variations of the real displacements • Equilibrium is a consequence of vanishing of a virtual work Internal Virtual Work External Virtual Work ετTdV Uf T B dV USTff f S dS UR iT i =+ +∑ C ∫∫ ∫ i VV Sf Stresses in equilibrium with applied loads Virtual strains corresponding to virtual displacements Method of Finite Elements I Institute of Structural Engineering Page 17 Principle of Complementary Virtual Work • The principle of virtual forces: virtual work of equilibrium variations of the stresses and the forces on the strains and displacements vanishes; the stress field considered is a statically admissible field of variation • Equilibrium is assumed to hold a priori and the compatibility of deformations is a consequence of vanishing of a virtual work • Both principles do not depend on a constitutive law Method of Finite Elements I Institute of Structural Engineering Page 18 Variational Formulation • Based on the principle of stationarity of a functional, which is usually potential or complementary energy • Two classes of boundary conditions: essential (geometric) and natural (force) boundary conditions • Scalar quantities (energies, potentials) are considered rather than vector quantities Method of Finite Elements I Institute of Structural Engineering Page 19 Principle of Minimum Total Potential Energy • Conservation of energy: Work = change in potential, kinetic and thermal energy • For elastic problems (linear and non-linear) a special case of the Principle of Virtual Work – Principle of minimum total potential energy can be applied • U is the stress potential: τεij=∂U / ∂ ij 1 U = ε T Cεdv ∫ 2 v T S W = ∫ f B Udv + ∫ f f Uds v s Method of Finite Elements I Institute of Structural Engineering Page 20 Principle of Minimum Total Potential Energy • Total potential energy is a sum of strain energy and potential of loads P = U −W • This equation, which gives P as a function of deformation components, together with compatibility relations within the solid and geometric boundary conditions, defines the so called Lagrange functional • Applying the variation we invoke the stationary condition of the functional P dP = dU − dW = 0 Method of Finite Elements I Institute of Structural Engineering Page 21 Variational Formulation • By utilizing the variational formulation, it is possible to obtain a formulation of the problem, which is of lower complexity than the original differential form (strong form). • This is also known as the weak form, which however can also be aained by following an alternate path (see Galerkin formulation). • For approximate solutions, a larger class of trial functions than in the differential formulation can be employed; for example, the trial functions need not satisfy the natural boundary conditions because these boundary conditions are implicitly contained in the functional – this is extensively used in MFE. Method of Finite Elements I Institute of Structural Engineering Page 22 Approximative Methods Instead of trying to find the exact solution of the continuous system, i.e., of the strong form, try to derive an estimate of what the solution should be at specific points within the system. The procedure of reducing the physical process to its discrete counterpart is the discretisation process. Method of Finite Elements I Institute of Structural Engineering Page 23 Approximative Methods Variational Methods Weighted Residual Methods approximation is based on the start with an estimate of the the solution and minimization of a functional, as those demand that its weighted average error is defned in the earlier slides.
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