The physics models
Chapter Outline
1.1 Heat flow fundamentals 2
1.1.1 Basic equations 2
1.1.2 Boundary conditions 2
1.1.3 Weak forms of the thermal equation 3
1.1.4 The shape functions for FEM 3
1.1.5 Formulations in matrix form 4
1.1.6 The nonlinearity in thermal analysis 5
1.1.7 Stabilization method for convection-dominant transport equations 5
1.1.8 Penalty-based thermal contact 5
1.2 Fluid dynamics 6
1.2.1 Basic equations for fluid flow 6
1.2.2 Boundary and initial conditions for fluid flow 7
1.2.3 The constitutive equation for fluid flow 8
1.2.4 The weak forms 8
1.2.5 Finite element equations 9
1.2.6 The nonlinearity and numerical challenging in CFD 12
1.2.7 The stabilization methods 12
1.2.8 Turbulence model in CFD 14
1.2.9 The general transport equations 23
1.3 Structural mechanics 26
1.3.1 Governing equations for structure analysis 26
1.3.2 The equation in matrix form 27
1.3.3 The general nonlinearity in structural dynamics 28
1.3.4 Incompressible hyperelastic material for the rubber (Watanabe, 1995) 29
1.3.5 Formulations for thin structure with large deformation 30
1.3.6 Mixed interpolation tensorial component shell element (Bathe, 2006) 32
1.3.7 Generalized conforming flat shell element 34
1.3.8 Contact problems in structural mechanics 46
1.4 Electromagnetic field 59
1.4.1 Fundamental equations 59
1.4.2 Classification of Maxwell’s equations and potential formulations 60
1.4.3 FEM discretization of potential formulations 65
1.4.4 Gauge methods for electromagnetic elements 80
1.4.5 Output results 83
1.4.6 Sliding interface coupling in electromagnetic problems 86
1.5 Acoustic analysis 93
1.5.1 Governing equations 93
1.5.2 Momentum equation 94
1.5.3 The boundary conditions 94
1.5.4 The equations in weak form 94
1.5.5 Acoustics Equations in Matrix Form 95
& S.
1.1
1.1.1
Heat flow fundamentals
Basic equations
The solution for heat flow analysis is to find the temperature distribution S () ∈ TXt , T , so that the following governing Equation (1.1) is satisfied.
The primary variable is temperature T in the infinite-dimensional space of ST. The first part on the left-hand side of Equation (1.1) is the time derivative term, the second term is the convection term caused by fluid flow, and the third term is the diffusion term. The source term of heat generation is on the right-hand side of Equation (1.1).
The material properties needed to be decided are; mass density, ρ; specific heat, cv; and thermal conductivity, k. For fully incompressible flow, ρ is assumed to be constant, but for slightly compressible and low-speed compressible flow, we assume ρ = ρ(p) and ρ = ρ( p, T), respectively. Also, we assume specific heat cv and conductivity k to be either constant or a function of temperature. v is the convective velocity, and qB is the heat generation from other physics models in multiphysics simulation.
Remark 1.1: qB is the source term of heat generation that may come from thermoelastic damping, fluid viscous heat, or electro- or electromagnetic heat in multiphysics simulation. On the other hand, temperature T as output may affect the material properties or other quantities of the corresponding physics models.
Symbol indicates the value that is received from other physics model, and represents the value transferred to other physics model in multiphysics simulation. Symbol means the value can be either sent out to or received from other physics model in multiphysics simulation.
1.1.2 Boundary conditions
Three types of boundary conditions are considered:
1. Specified temperature on Γg:
Here, Ts is the specified temperature.
2. Specified heat flow on Γ
where, q s is the specified heat flow and ni is the component of the unit normal direction n.
Remark 1.2: The boundary conditions: Equations (1.2) and (1.3) can be used in conjugate heat transfer coupling.
3. Specified convection surfaces acting over surface Γq2:
Where; h f , is the convective heat transfer coefficient; TB , is the bulk temperature of the adjacent physics model; and TS , is the temperature at the surface of the model.
Remark 1.3: The positive specified heat flow is into the boundary.
1.1.3 Weak forms of the thermal equation
The weak form of heat flow equations is given in Equation (1.5) as follows: find temperature S ∈ TT , such that W ∀∈ wTT :
Here, wT is the test function of temperature T in infinite-dimensional space WT. The semi-discrete Galerkin finite element formulation of the heat flow problem is stated as: find S ∈ ThTh , such that W ∀∈ wT h T h :
Where, ST h and WT h are the sets of finite-dimensional trial and test functions for temperature, respectively.
1.1.4 The shape functions for FEM
The basic shape function for the coordinate:
Here; ξ () N i E , is the shape function at node i; ξ, is the element local coordinate; and xi, is the nodal coordinate at node i; nen, is the number of nodes per element, respectively. The interpolation for temperature variables:
Here, N i T is the shape function for temperature at node i, which is consistent with N i E for the class of isoparametrical element. The standard isoparametrical element can be used for the temperature interpolation, for example, the low order 4-node tetrahedral element and 8-node hexahedral element, are the most commonly used for thermal analysis, although the high order 10-node tetrahedral element and 20-node hexahedral element are also useful in achieving better accuracy with the same number of nodes. The other shapes, for example, wedge and pyramid can be used for the transition region between tetrahedral and hexahedral elements. The details about these shape functions can be found in Bathe (2006) and Wang (2000).
1.1.5 Formulations in matrix form
Integrating of Equation (1.6) by part for the third term on the left-hand side, applying boundary conditions of Equations (1.2)–(1.4), and substituting the shape functions into it, one can obtain assembled global Equation (1.9) in matrix form:
= = CC e ne e 1 , ∫ ρ =Ω ν Ω CcNN d eiji T j T (, ) e , is the element specific heat matrix. ∑ = = KK tm e ne e tm 1 , ∫ ρν =Ω ν Ω KcNN d eij tm i T kjk T (, ), e is the element mass transport conductivity matrix.
∑ = = KK tb e ne e tb 1 , ∫ =Ω Ω KNkN d eij tb im T mmjm T (, ), , e is the element diffusion conductivity matrix.
∑ = = KK tc e ne e tc 1 , ∫ =Γ Γ KhNN d eij tc fi T j T (, ) q 2 is element convection surface conductivity matrix.
∑ = QQ f f flux flux , ∫ =Γ Γ QNq d fii Ts () flux q 1 is element surface heat flux vector.
∑ = QQ f f conv conv , ∫ =Γ Γ QThN d fiBfi T () conv q 2 is the element convection surface heat flow vector.
∑ = QQ g e e g , ∫ =Ω Ω QqN d ei gB i T () e is the element heat generation vector. Here, ∑ = e ne 1 represents the assembly operator for element matrix or vector with number of ne elements.
∑ f represents the assembly operator for element surface loads.
1.1.6 The nonlinearity in thermal analysis
1.1.6.1 Material properties
The material properties depend on the temperature, that is, == νν kkTccT (),( )
1.1.6.2 Convection term from computational fluid dynamics (CFD) coupling
The definition of Peclet number for convective heat transfer problem is
Where, he stands for element size.
For a heat flow problem with Pe ≥ 2.0, to avoid oscillation of the solution of the temperature field and to get a better quality matrix equation, the spatial stabilization method may be needed. The streamline upwind Petrov/Galerkin (SUPG) method is presented in the further section.
1.1.7 Stabilization method for convection-dominant transport equations
The SUPG method is used for the convection-dominated heat transfer problem for the case, where Peclet number is higher than 2.0. For details about implementation of the stabilization method for transport equations, please refer Section 1.2.9.3.
1.1.8 Penalty-based thermal contact
The thermal contact problem can be used for thin air gap or other thin-layered problems with different thermal conductivities.
Assuming the thickness of a thin air gap is d, and T1 and T2 are the temperatures on the side A and side B of the contact wall, respectively, then the heat flux from T2 side into T1 side is:
Here, k is the thermal conductivity, and the heat flux from T1 side into T2 side is the negative of q in Equation (1.11). Here we assume the air layer is thin enough, so linear distribution assumption of temperature across the air gap is acceptable.
Then thermal conductance for this gap is: = h k fd (1.12)
where h f , as the contact thermal conductance, can be setup as input property.
1.1.8.1 The matrix equation for thermal contact
The heat flow for side A
The previous equation can be added into the left-hand side of Equation (1.5) to consider the thermal contact affect from the air gap, and the matrix form can be expressed as:
where, Ai is the surface node i on side A and Bj is the surface node j on side B, respectively. + kTkT AiAjAAiBjB needs to be added into the left side of Equation (1.9) for side A, and its negative value will be added in the same way for side B.
1.2 Fluid dynamics
1.2.1 Basic equations for
fluid flow
We assume the viscous fluid to be isothermal and barotropic (i.e., ρ () = Fp,0
and that ∂∂ρρ = pB // , in which B , p , and ρ are fluid bulk modulus, pressure, and fluid density, respectively. The Arbitrary Lagrangian Eulerian (ALE) formulation is usually used to handle moving boundary problems of fluid flow in coupling analysis. The fundamental equations of fluid flow (Zhang and Hisada, 2001) are expressed as:
The primary variables are pressure p, velocity vector v, and temperature T. Here, σ is the Cauchy stress tensor, g is the acceleration of gravity, and =−cv vm is the relative velocity of fluid particles to the ALE coordinates with vm the mesh velocity. Ωt f denotes the spatial thermal fluid domain bounded by the boundary Γ t f of interest at any instant t. Here the superscript f stands for the fluid component. l is the second viscosity, and k is the thermal conductivity. And in Equation (1.18), the strain rate related energy term is expressed as:
== + Deeevv with 1 2 ().ijijijijji 2 ,,
Remark 1.4: The source term fi in Equation (1.17) may come from the coupling physics, for example, the electromagnetic force. The outgoing variables are the pressure p and velocity vector v in the coupled physics simulation.
1.2.2 Boundary and initial conditions for fluid flow
The boundary is composed of Γ t g and Γ t h corresponding to Dirichlet- and Neumanntype boundary conditions, respectively.
And subject to the following initial conditions:
The boundary conditions for the energy equation are the same as those in previous sections.
Remark 1.5: Equations (1.19) and (1.20) represent the coupling boundaries in the fluid–structure interaction (FSI) problem with gi received from the structure and hi sent to the structure.
1.2.3 The constitutive equation for fluid flow
The fluid is assumed to be Newtonian, and the constitutive equation is:
where dij is the component of identity tensor and m is the dynamic viscosity of the fluid. The properties for thermal are provided in the previous section.
1.2.4 The weak forms
1.2.4.1 Galerkin formulation for N–S equations
The weak forms for the fluid equations given in Equations (1.16)–(1.18) by the Galerkin method are written as: find pressure S ∈ pp , velocity S ∈ v v , and temperature S ∈ TT , such that WW W ∀∈ ∀∈ ∀∈ w ww ,, and ppvvTT :
Where Sp, Sv, and ST denote the sets of infinite-dimensional trial functions for pressure, velocity, and temperature, respectively. W p, W v, and W T are the sets of test functions (weighing functions) for the continuity, equilibrium, and energy equations, respectively.
The Galerkin finite element formulation of the fluid equations is stated as follows: find pressure
and
1.2.4.1.1 The shape functions
Basic shape function for the coordinate:
Shape function for any solution variables:
Here, N a v and N a p are the shape function for velocity and pressure at node a , respectively. The shape function for velocity and pressure can be consistent or different, and the pressure stabilized Petrov–Galerkin (PSPG) stabilization algorithm will be needed to avoid the pressure oscillation if the pressure has the same shape function as the velocity.
1.2.5
Finite element equations
By using the finite element discretization for the ALE form of Navier–Stokes Equations (1.30)–(1.32), we can obtain the finite element Equation (1.36).
MP and M are the generalized mass matrices for pressure and velocity, respectively. ΛΛP and ΛΛ are the generalized matrices of convective terms for pressure and velocity, respectively. Km is the fluid viscosity matrix, G is the divergence operator matrix. pf and Vf are the pressure and velocity vectors, respectively. Ff is the external force vector that consist of the directly applied external load vector F v E and the coupling load vector from electromagnetic field FEM v Qf(p, v) is the heat flow vector caused by viscous flow. Q(qB) consists of other coupling terms of heat flow vector contributed by the thermoelastic damping or electromagnetic heat. QE is the heat flow vector contributed by any other directly applied external heat boundaries.* indicates the time derivative in the ALE coordinates. In the present work, the secant matrices are used for nonlinear iterations, so the incremental forms of Equation (1.36) can be simply obtained.
For the continuity equation:
Here, nep, number of pressure nodes per element; nev, number of velocity nodes per element; and NSD, spatial dimension of the analysis domain.
For equilibrium equations:
Remark 1.6: The formulations and engineering applications of the finite volume method for compressible fluid flow are given in Chapter 10.
1.2.6 The nonlinearity and numerical challenging in CFD
The nonlinear term in Navier–Stokes equations of Equation (1.17) is the convection term, and most of the numerical difficulties and stability issues for fluid flow are caused by this term.
The numerical oscillation of the solutions in space for the convection-dominated flow may happen because of the usage of an inappropriate interpolation scheme. For a high Peclet number problem (Pe > 2), the central differencing scheme usually produces numerical oscillations in space, and the upwind scheme may help to prevent from oscillation but with a compromise on the numerical accuracy.
For the fluid flow with a high Reynolds number, the flow can be turbulence with multiscale responses. Extreme fine mesh discretization must be needed for resolving the finescale flow phenomenal when using the direct numerical method. However, it is usually unpractical for huge computational requests. The turbulence model, for example, k-epsilon model, large eddy simulation [LES] model can be used for modeling fine-scale effects on the coarse scale mesh discretization level. Details about the stabilization method are given in the Section 1.2.7, and the turbulence models are presented in Section 1.2.8.
1.2.7 The stabilization methods
1.2.7.1 SUPG and PSPG methods
The SUPG and PSPG formulations for Navier–Stokes equations are stated as find pressure
Here the residual vector of the momentum equation is:
The last two terms in Equation(1.54) are SUPG and PSPG stabilization terms, respectively.
The commonly used definitions of τSUPG and τPSPG = τSUPG are given by Bazilevs et al. (2013):
Here, hr is the generalized element size,
and j is the unit velocity of the gradient of the norm of the convective velocity,
1.2.7.2 Discontinuity capturing operator (Tezduyard, 2012)
The additional term to overcome the undershoot and overshoot problem in convection dominant flow is the discontinuity capturing operator:
If the Y, Z, b method is used (Tezduyar and Senga, 2007),
Some details about the stabilization method are presented in Chapter 5.
1.2.7.3 Underrelaxation method and solution capping
The underrelaxation scheme and solution capping schemes are convenient and useful for stabilizing the solution during the nonlinear iteration process when facing inconsistent convergence or even divergence phenomena. The underrelaxation methods can be used for updating the major primary variable, for example, velocity, pressure, temperature. Details about the underrelaxation scheme and solution capping scheme are presented in Sections 5.4 and 5.5, respectively.
1.2.8 Turbulence model in CFD
Turbulence causes eddies of flow with a wide range of spatial length and time scales that interact in a dynamically complex way. The research about turbulence model is one of the major subjects for development of CFD. The ways to solve turbulence flow can be classified into three categories: turbulence modes for Reynolds-averaged Navier–Stokes (RANS) equation, LES, and direct numerical simulation (DNS).
1.2.8.1 k-Epsilon turbulence model
For the RANS model, the effects of the turbulence on the time-averaged mean flow are modeled with classical turbulence models. The most popular ones, the k-epsilon and k-omega models, are presented in this section. Compared with LES and DNS, the computing resources needed for reasonable predictions of flow field are modest. Two extra transport equations need to be solved for the k-epsilon and k-omega models.
1.2.8.1.1 Basic equations for the k-epsilon model
The governing equation for turbulence kinetic energy k :
The similar transport equation for the rate of viscous dissipation ε:
The turbulence production term attributable to viscous forces is,
If the flow is incompressible, then the second term is vanishing and µ ⋅⋅ = PSS2. ktijij
The effective viscosity
ν =+ , t eff (1.69) where the eddy viscosity, ν ε = µC k . t 2 (1.70)
The parameters Cm, Cε1, Cε2, σk, and σε are constant values, the standard k-epsilon model takes the value: Cm = 0.09, Cε1 = 1.44, Cε2 = 1.92, σk = 1.0, and σε = 1.3.
Rewriting Equations (1.65) and (1.66), we have:
1.2.8.1.2 Equations in weak form
Integration by part for third term on the left of Equation (1.73):
By divergence theorem, we have
where n is the normal of the boundary.
1.2.8.1.3
Boundary conditions
The boundary types for turbulence term consist of the inlet, outlet, free stream, and solid walls.
Inlet: distribution of k and ε must be given.
Outlet, symmetry: ∂ ∂ = k n 0 , ∂ε ∂ = n 0 ;
Free stream: distribution of k and ε must be given as ∂ ∂ = k n 0 and ∂ε ∂ = n 0 ;
Solid wall: approach depends on Reynolds number.
The boundary layer consists of near wall viscous sublayer, log law layer, turbulence region, and inertial dominated region. To have a good resolution with reasonable mesh, the wall function for the logarithmic layer is normally used.
The logarithmic relation for the near wall velocity is given by:
Here, dimensionless wall distance:
Now, + v is the near wall velocity; Vt is the known velocity tangent to the wall at distance of ∆y from the wall; τw is the wall shear stress; k is the von Karman constant; and E is law of the wall constant. The value of k and E are 0.4 and 0.9, respectively, and the latter corresponds to a smooth wall condition.
Based on the known value of Vt and the near wall distance ∆y, by using the log law equation, the wall shear stress τw can be obtained, and so is the calculation of dynamic viscosity:
The wall element viscosity value is the larger one of the laminar viscosity and the calculated value from the Equation (1.80)
The adjacent value of the turbulence kinetic energy is obtained from the k-epsilon model. The near wall value of dissipation rate is dominated by the length scale and is given by equation:
1.2.8.1.4 Equations in matrix form
The shape functions of k and ε:
Substituting Equations (1.82) and (1.83) into the weak form of the k-epsilon Equations (1.73) and (1.74), one has:
Where the matrices and vectors in Equations (1.84) and (1.85) can be obtained by a standard assembly process of the element matrices and vectors. The components in each matrix or vector at element level are given further.
For k equation at element level:
For epsilon equation at element level:
1.2.8.2 Wilcox k-omega turbulence model
1.2.8.2.1 Basic equations for k-omega model
The transport equation for k and w for turbulence flow:
Here, b1, β* , g1, σk, and σw are the constants, b1 = 0.075, β* = 0.09, g1 = 0.553, σk = 2.0, and σw = 2.0. If we use this variable, the length scale is ω = k / , and the eddy dynamic viscosity:
The Reynolds stress
1.2.8.2.2 Boundary conditions
Inlet: distribution of k and w must be given.
Outlet: symmetry:
= ∂
= k nn 0, 0; Free stream: small none zero w must be specified.
Solid wall: in boundary conditions on solid wall for k, the k-omega model of Wilcox has the advantage that an analytical expression is known for w in the viscous sublayer. The main idea behind the present formulation is to blend the wall value of w between the logarithmic and near wall formulation.
The algebraic expression for w in the logarithmic region:
And the corresponding expression in the viscous sublayer:
where ∆y is the distance between the first and the second mesh point. To achieve smooth blending and to avoid cyclic convergence behavior, the following formation is adopted.
1.2.8.2.3
Weak forms of k-omega model
The weak forms for k w can be obtained as the procedure for k ε equation, as well as for the finite element method (FEM) equations in matrix form.
1.2.8.2.4 Equations in matrix form
The shape functions for k and ω :
The assembled FEM equations:
Where for k equation at element level:
For the omega equation at element level:
1.2.8.3 Procedure for solving the k-epsilon/k-omega turbulence model
Time loop
PMA predictor
Iteration loop
Solve N–S equations
Solve k-epsilon equation
Solve k equation
Solve epsilon/omega equation
Loop till k and epsilon/omega equation converge
PMA corrector
Solve mesh motion
End of iteration loop
End of time loop
1.2.8.4 Large eddy simulation
Instead of time averaging, LES uses a spatial filtering operation to separate the larger and smaller eddies. The effects of unresolved small eddies on the mesh level are included by means of subgrid scale (SGS) models. Transient flow equations must be solved, so much larger demands on computer storage and computing speed are requested compared with the RANS model.
SpatialfilteringofunsteadyNavier–Stokeequations: in LES, we define a spatial filtering operation by means of a filter function G(x, x ′ , ∆)
where
φ () x t , = filtered function
φ() x t , = original (unfiltered) function
∆ = filter cutoff width
In this section, the over bar indicates that the spatial filtering is not time averaging. The choices for the filtering (Versteeg and Malalasekera, 2007):
• Box filter:
• Gaussian filter:
g is the parameter, and the typical value of it is 6.0.
• Spectral cutoff
The Box filter is normally used in finite volume implementations of the LES method. The Gaussian filter is used in finite element and finite differences implementation. The spectral cutoff filter gives a sharp cutoff in the energy spectrum at the wavelength of ∆/π, but it cannot be used in general-purpose CFD.
The filter cutoff width can be defined as
for three-dimensional meshes. Here, ∆x, ∆y, ∆z are the element sizes in x, y, z directions, respectively.
