Introduction Description of the method Simple example General features Ritz Method Introductory Course on Multiphysics Modelling T OMASZ G. Z IELI ´ NSKI bluebox.ippt.pan.pl/˜tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences Warsaw • Poland
Introduction Description of the method Simple example General features Outline Introduction 1 Direct variational methods Mathematical preliminaries
Introduction Description of the method Simple example General features Outline Introduction 1 Direct variational methods Mathematical preliminaries Description of the method 2 Basic idea Ritz equations for the parameters Properties of approximation functions
Introduction Description of the method Simple example General features Outline Introduction 1 Direct variational methods Mathematical preliminaries Description of the method 2 Basic idea Ritz equations for the parameters Properties of approximation functions Simple example 3 Problem definition Variational statement of the problem Problem approximation and solution
Introduction Description of the method Simple example General features Outline Introduction 1 Direct variational methods Mathematical preliminaries Description of the method 2 Basic idea Ritz equations for the parameters Properties of approximation functions Simple example 3 Problem definition Variational statement of the problem Problem approximation and solution General features 4
Introduction Description of the method Simple example General features Outline Introduction 1 Direct variational methods Mathematical preliminaries Description of the method 2 Basic idea Ritz equations for the parameters Properties of approximation functions Simple example 3 Problem definition Variational statement of the problem Problem approximation and solution General features 4
Introduction Description of the method Simple example General features Direct variational methods Direct methods – the methods which (bypassing the derivation of the Euler equations) go directly from a variational statement of the problem to the solution.
Introduction Description of the method Simple example General features Direct variational methods Direct methods – the methods which (bypassing the derivation of the Euler equations) go directly from a variational statement of the problem to the solution. The assumed solutions in the variational methods are in the form of a finite linear combination of undetermined parameters with appropriately chosen functions . In these methods a continuous function is represented by a finite linear combination of functions . However, in general, the solution of a continuum problem cannot be represented by a finite set of functions an error is introduced into the solution. The solution obtained is an approximation of the true solution for the equations describing a physical problem. As the number of linearly independent terms in the assumed solution is increased , the error in the approximation will be reduced (the solution converges to the desired solution)
Introduction Description of the method Simple example General features Direct variational methods Direct methods – the methods which (bypassing the derivation of the Euler equations) go directly from a variational statement of the problem to the solution. The assumed solutions in the variational methods are in the form of a finite linear combination of undetermined parameters with appropriately chosen functions . In these methods a continuous function is represented by a finite linear combination of functions . However, in general, the solution of a continuum problem cannot be represented by a finite set of functions an error is introduced into the solution. The solution obtained is an approximation of the true solution for the equations describing a physical problem. As the number of linearly independent terms in the assumed solution is increased , the error in the approximation will be reduced (the solution converges to the desired solution) Classical variational methods of approximation are: Ritz , Galerikin , Petrov-Galerkin (weighted residuals).
Introduction Description of the method Simple example General features Mathematical preliminaries Theorem (Uniqueness) If A is a strictly positive operator (i.e., � A u , u � H > 0 holds for all 0 � = u ∈ D A , and � A u , u � H = 0 if and only if u = 0 ), then A u = f in H has at most one solution ¯ u ∈ D A in H . SKIP PROOF
Introduction Description of the method Simple example General features Mathematical preliminaries Theorem (Uniqueness) If A is a strictly positive operator (i.e., � A u , u � H > 0 holds for all 0 � = u ∈ D A , and � A u , u � H = 0 if and only if u = 0 ), then A u = f in H has at most one solution ¯ u ∈ D A in H . Proof. Suppose that there exist two solutions ¯ u 1 , ¯ u 2 ∈ D A . Then � � A ¯ u 1 = f and A ¯ u 2 = f → A ¯ u 1 − ¯ u 2 = 0 in H , and � � � � ¯ u 1 − ¯ , ¯ u 1 − ¯ H = 0 → ¯ u 1 − ¯ u 2 = 0 ¯ u 1 = ¯ A u 2 u 2 or u 2 . QED
Introduction Description of the method Simple example General features Mathematical preliminaries Theorem A : D A → H be a positive operator (in D A ), and f ∈ H ; Π : D A → H be a quadratic functional defined as Π( u ) = 1 2 � A u , u � H − � f , u � H .
Introduction Description of the method Simple example General features Mathematical preliminaries Theorem A : D A → H be a positive operator (in D A ), and f ∈ H ; Π : D A → H be a quadratic functional defined as Π( u ) = 1 2 � A u , u � H − � f , u � H . 1 If ¯ u ∈ D A is a solution to the operator equation A u = f in H , then the quadratic functional Π( u ) assumes its minimal value in D A for the element ¯ u , i.e., Π( u ) ≥ Π(¯ u ) and Π( u ) = Π(¯ u ) only for u = ¯ u .
Introduction Description of the method Simple example General features Mathematical preliminaries Theorem A : D A → H be a positive operator (in D A ), and f ∈ H ; Π : D A → H be a quadratic functional defined as Π( u ) = 1 2 � A u , u � H − � f , u � H . 1 If ¯ u ∈ D A is a solution to the operator equation A u = f in H , then the quadratic functional Π( u ) assumes its minimal value in D A for the element ¯ u , i.e., Π( u ) ≥ Π(¯ u ) and Π( u ) = Π(¯ u ) only for u = ¯ u . 2 Conversely, if Π( u ) assumes its minimal value, among all u ∈ D A , for the element ¯ u , then ¯ u is the solution of the operator equation, that is, A ¯ u = f .
Introduction Description of the method Simple example General features Mathematical preliminaries Example: a self-adjoint operator � � Let: u , v ∈ H = all differentiable functions on [ 0 , L ] , α = α ( x ) . L � The inner (scalar) product in H is defined as: � u , v � H ≡ u v d x . 0 A ( u ) ≡ d � α d u � The linear mapping A : H → H , is defined as: . d x d x
Introduction Description of the method Simple example General features Mathematical preliminaries Example: a self-adjoint operator � � Let: u , v ∈ H = all differentiable functions on [ 0 , L ] , α = α ( x ) . L � The inner (scalar) product in H is defined as: � u , v � H ≡ u v d x . 0 A ( u ) ≡ d � α d u � The linear mapping A : H → H , is defined as: . d x d x This is a self-adjoint operator , namely: L L � � � − d � α d u �� � � � A u , v � H = A u v d x = v d x d x d x 0 0 L � d v L � L � � − α d u � α d u � α d u d v = d x v + d x d x = d x d x d x d x 0 0 0 L � L � α d v � u d � α d v � = d x u − d x d x d x 0 0 L L � u d � − α d v � � � � d x = � u , A v � H = d x = u A v d x d x 0 0
Introduction Description of the method Simple example General features Outline Introduction 1 Direct variational methods Mathematical preliminaries Description of the method 2 Basic idea Ritz equations for the parameters Properties of approximation functions Simple example 3 Problem definition Variational statement of the problem Problem approximation and solution General features 4
Introduction Description of the method Simple example General features Basic idea 1 The problem must be stated in a variational form, as a minimization problem , that is: find ¯ u minimizing certain functional Π( u ) .
Introduction Description of the method Simple example General features Basic idea 1 The problem must be stated in a variational form, as a minimization problem , that is: find ¯ u minimizing certain functional Π( u ) . 2 The solution is approximated by a finite linear combination of the following form N � ¯ u ( x ) ≈ ˜ ( N ) ( x ) = c j φ j ( x ) + φ 0 ( x ) , u j = 1 where: c j denote the undetermined parameters termed the Ritz coefficients , φ 0 , φ j are the approximation functions ( j = 1 , . . . , N ).
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