Chapter5 - Chapter 5 Finite Element Formulation for Scalar...

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Chapter 5 1 Chapter 5 Finite Element Formulation for Scalar Field Problems 1. Introduction A 2-D scalar field problem has one single unknown dependent function u(x,y) . For example, u(x,y) may represent the temperature distribution in heat conduction problem, or the stress function in torsion of shaft, or the piezometric head in groundwater flow problem. Scalar field problems are, therefore, mathematically simpler than elasticity problems. Solving a scalar field problem means solving a partial differential equation (of one dependent variable) subject to certain boundary conditions. The systems of coupled partial differential equations, in the elasticity problems that we have solved, look a lot more formidable than most scalar field equations. For example, the relatively simple plane-stress/strain problem in Chapter 2 is governed by the following two equations: x c 11 u x + c 12 v y y c 66 u y + v x = f x x c 66 u y + v x y c 12 u x + c 22 v y = f y (1.1) with the boundary conditions: c 11 u x + c 12 v y n x + c 66 u y + v x n y = t x c 66 u y + v x n x + c 12 u x + c 22 v y n y = t y on S t v = v v = u } on S u (1.2) And yet, we run circle around them by resorting to principles such as the total potential energy principle or the virtual work principle. These principles help: • to formulate the element matrices, and • to set up the system algebraic equations. With regard to scalar field problems, a conceptual difficulty exists due to the apparent lack of similar physical principles. This difficulty is resolved by working backward from the governing partial differential equation to arrive at something resembling the total potential
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Chapter 5 2 principle or the virtual work principle, and from then on, the standard FE machinery will take over. 2. Weak Form of PDE for Boundary Value Problems We consider the following general form of a partial differential equation that applies to a variety of physical problems, three of which have been mentioned in the opening paragraph. Solve for u(x,y) of the second order PDE: A ( u ) ≡− ∂σ x ( u ) x ∂σ y ( u ) y + c o u f = 0 (2.1) where σ x ( u ) = c 11 u x + c 12 u y σ y ( u ) = c 21 u x + c 22 u y (2.2) and the coefficients c 's and f are given data. The boundary conditions will be specified subsequently. The weak form is an equivalent statement of the same problem. It resembles the virtual work principle or the total potential principle. To develop the weak form of Eq.2.1, we argue that since A(u) (Eq.2.1) has to be zero at every point in the domain V , it follows that V wA ( u ) dV = 0 (2.3) where w is an arbitrary function. Conversely, if Eq.2.3 is satisfied for all w , then Eq.2.1 is true at all points in V . To prove the statement, suppose A(u)
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This note was uploaded on 02/08/2010 for the course MECHANICAL 6371 taught by Professor Ha during the Winter '10 term at Concordia Canada.

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Chapter5 - Chapter 5 Finite Element Formulation for Scalar...

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