Notation remarks Local differential formulation Global integral formulations Matrix formulations Galerkin Finite Element Model for Heat Transfer 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
Notation remarks Local differential formulation Global integral formulations Matrix formulations Outline Notation remarks 1
Notation remarks Local differential formulation Global integral formulations Matrix formulations Outline Notation remarks 1 Local differential formulation 2 Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem
Notation remarks Local differential formulation Global integral formulations Matrix formulations Outline Notation remarks 1 Local differential formulation 2 Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem Global integral formulations 3 Test functions Weighted formulation and weak variational form
Notation remarks Local differential formulation Global integral formulations Matrix formulations Outline Notation remarks 1 Local differential formulation 2 Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem Global integral formulations 3 Test functions Weighted formulation and weak variational form Matrix formulations 4 Approximation Transient heat transfer (ordinary differential equations) Stationary heat transfer (algebraic equations)
Notation remarks Local differential formulation Global integral formulations Matrix formulations Outline Notation remarks 1 Local differential formulation 2 Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem Global integral formulations 3 Test functions Weighted formulation and weak variational form Matrix formulations 4 Approximation Transient heat transfer (ordinary differential equations) Stationary heat transfer (algebraic equations)
Notation remarks Local differential formulation Global integral formulations Matrix formulations Notation remarks The index notation is used with summation over the index i . Consequently, the summation rule is also applied for the approximation expressions, that is, over the indices r , s = 1 , . . . N (where N is the number of degrees of freedom). The symbol ( . . . ) | i means a (generalized) invariant partial differentiation over the i -th coordinate: ( . . . ) | i = ∂ ( . . . ) . ∂ x i The invariance involves the so-called Christoffel symbols (in the case of curvilinear systems of reference). Symbols d V and d S are completely omitted in all the integrals presented below since it is obvious that one integrates over the specified domain or boundary. Therefore, one should understand that: � � � � ( . . . ) = ( . . . ) d V ( x ) , ( . . . ) = ( . . . ) d S ( x ) . B B ∂ B ∂ B
Notation remarks Local differential formulation Global integral formulations Matrix formulations Outline Notation remarks 1 Local differential formulation 2 Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem Global integral formulations 3 Test functions Weighted formulation and weak variational form Matrix formulations 4 Approximation Transient heat transfer (ordinary differential equations) Stationary heat transfer (algebraic equations)
Notation remarks Local differential formulation Global integral formulations Matrix formulations PDE for Heat Transfer Problem ∂ T ∂ x · n > 0 (warm) Material data: ∂ T � kg ∂ x · n < 0 � n ̺ = ̺ ( x ) – the density m 3 (cold) � � J c = c ( x ) – the thermal capacity kg · K � W � ∂ B k = k ( x ) – the thermal conductivity m · K B heat sink Known fields: � W ( ̺, c , k ) � f = f ( x , t ) – the heat production rate f < 0 m 3 � m � u i = u i ( x , t ) – the convective velocity s heat source f > 0 The unknown field: � � T = T ( x , t ) =? – the temperature K
Notation remarks Local differential formulation Global integral formulations Matrix formulations PDE for Heat Transfer Problem ∂ T ∂ x · n > 0 (warm) Material data: ∂ T � kg ∂ x · n < 0 � n ̺ = ̺ ( x ) – the density m 3 (cold) � � J c = c ( x ) – the thermal capacity kg · K � W � ∂ B k = k ( x ) – the thermal conductivity m · K B heat sink Known fields: � W ( ̺, c , k ) � f = f ( x , t ) – the heat production rate f < 0 m 3 � m � u i = u i ( x , t ) – the convective velocity s heat source f > 0 The unknown field: � � T = T ( x , t ) =? – the temperature K Heat transfer equation � W � • ̺ c T + q i | i − f = 0 where the heat flux vector : K � − k T | i – for conduction (only), q i = q i ( T ) = − k T | i + ̺ c u i T – for conduction and convection, � K � • T = ∂ T and ∂ t is the time rate of change of temperature . s
Notation remarks Local differential formulation Global integral formulations Matrix formulations Initial and boundary conditions The initial condition (at t = t 0 ) T ( x , t 0 ) = T 0 ( x ) in B Prescribed field : � � T 0 = T 0 ( x ) – the initial temperature K
Notation remarks Local differential formulation Global integral formulations Matrix formulations Initial and boundary conditions The initial condition (at t = t 0 ) T ( x , t 0 ) = T 0 ( x ) in B Prescribed field : ∂ T ∂ x · n > 0 � � T 0 = T 0 ( x ) – the initial temperature K (warm) ∂ T ∂ x · n < 0 n (cold) The boundary conditions (on ∂ B ) ∂ B the Dirichlet type: B T ( x , t ) = ˆ heat sink T ( x , t ) on ∂ B T ( ̺, c , k ) f < 0 the Neumann type: heat source � � − q i T ( x , t ) n i = ˆ q ( x , t ) on ∂ B q f > 0 Prescribed fields : T = ˆ ˆ � � T ( x , t ) – the temperature K � W � q = ˆ ˆ q ( x , t ) – the inward heat flux m 2
Notation remarks Local differential formulation Global integral formulations Matrix formulations Initial-Boundary-Value Problem IBVP of the heat transfer Find T = T ( x , t ) for x ∈ B and t ∈ [ t 0 , t 1 ] satisfying the equation of heat transfer by conduction (a), or by conduction and convection (b): � − k T | i ← (a) • T + q i | i − f = 0 where q i = q i ( T ) = ̺ c − k T | i + ̺ c u i T ← (b) with the initial condition (at t = t 0 ): T ( x , t 0 ) = T 0 ( x ) in B , and subject to the boundary conditions : T ( x , t ) = ˆ − q i � � n i = ˆ T ( x , t ) on ∂ B T , T ( x , t ) q ( x , t ) on ∂ B q , where ∂ B T ∪ ∂ B q = ∂ B and ∂ B T ∩ ∂ B q = ∅ .
Notation remarks Local differential formulation Global integral formulations Matrix formulations Outline Notation remarks 1 Local differential formulation 2 Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem Global integral formulations 3 Test functions Weighted formulation and weak variational form Matrix formulations 4 Approximation Transient heat transfer (ordinary differential equations) Stationary heat transfer (algebraic equations)
Notation remarks Local differential formulation Global integral formulations Matrix formulations Test functions T ( x , t ) , δ T ( x ) solution and trial functions, T ˆ T test functions, δ T x ∂ B T ∂ B q Dirichlet b.c. Neumann b.c. T = ˆ T , δ T = 0 Test function δ T ( x ) is an arbitrary (but sufficiently regular) function defined in B , which meets the admissibility condition : δ T = 0 on ∂ B T . Notice that test functions are always time-independent .
Notation remarks Local differential formulation Global integral formulations Matrix formulations Weighted formulation and weak variational form Weighted integral formulation ✎ ☞ � � � • ̺ c T + q i | i − f δ T = 0 (for every δ T ) ✍ B ✌
Notation remarks Local differential formulation Global integral formulations Matrix formulations Weighted formulation and weak variational form Weighted integral formulation ✎ ☞ � � � • ̺ c T δ T + q i | i δ T − f δ T = 0 (for every δ T ) ✍ B B B ✌ The term q i | i introduces the second derivative of T : q i | i = − k T | ii + . . . . However, the heat PDE needs to be satisfied in the integral sense. Therefore, the requirements for T can be weaken as follows.
Notation remarks Local differential formulation Global integral formulations Matrix formulations Weighted formulation and weak variational form Weighted integral formulation ✎ ☞ � � � • ̺ c T δ T + q i | i δ T − f δ T = 0 (for every δ T ) ✍ B B B ✌ The term q i | i introduces the second derivative of T : q i | i = − k T | ii + . . . . However, the heat PDE needs to be satisfied in the integral sense. Therefore, the requirements for T can be weaken as follows. Integrating by parts (using the divergence theorem) � � � � � q i | i δ T = ( q i δ T ) | i − q i δ T | i = q i δ T n i − q i δ T | i B B B ∂ B B
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