Control Theory Approach to Aero Shape Optimization

Transcription

1 Control Theory Approach to Aero Shape Optimization. Overview. Finite Volume Method 3. Formulation of Optimal Design Problem 4. Continuous Adjoint Approach 5. Discrete Adjoint Approach

2 Overview of Adjoint Method Step : Define Cost Function and Flow Field Equation I = I(w, F) R(w, F) = 0 w F = flow field variables = design variables Step : Variation of Cost Function and Flow Field Equation δi = I T w δi = R w δw I F T δw R F δf δf

3 Overview of Adjoint Method Step 3: Multiply the variation of the Flow Field Equation with a Lagrange Multiplier and introduce it as a constraint. δi = I w [ = T T I w δw I F T ψ T R w [ ] R δf ψ T R δw w F δf ] [ ] T I δw ψ T R δf w F Step 4: Choose ψ such that it satisfies the Adjoint Equation [ R w ] T ψ = I w

4 Overview of Adjoint Method 3 Step 5: Calculate Gradient δi = [ T I w ψ T R F ] δf = G T δf Gradient does not contain terms that are a function of the variation of the flow field variables

5 Overview of Adjoint Method 4 Field Equations Continuous Adjoint Equations Discrete Field Equations Discretized Continuous Adjoint Equations Discrete Adjoint Equations

6 Overview of Adjoint Method 5 Continuous Adjoint Approach Control Theory applied to differential equations governing the flow. The variation of the flow field equation is introduced as a constraint into the design problem through the use of Lagrange multipliers. Collect terms associated with the variation of the flow field variables to produce the continuous adjoint equations and its boundary conditions. The terms associated with the variation of the design variables produce the gradient. Advantages: Provides an analytical form of the equations. Stability analysis and analytical solutions can be developed to understand the characteristics and behavior of the equations Disadvantage: Provides the inexact gradient to the exact cost function.

7 Overview of Adjoint Method 6 Discrete Adjoint Approach Control Theory applied to discrete field equations. The variation of the discrete flow field equation is introduced as a constraint into the design problem through the use of Lagrange multipliers. Collect terms multiplied by the variation δw i of the discrete flow field variables to produce the discrete adjoint equations and its boundary conditions. Advantages: Provides the exact gradient to the inexact cost function. Boundary conditions are consistent with the numerical scheme. Disadvantage: For explicit type schemes, development cost is high. Limiters are not differentiable.

8 Overview of Adjoint Method 7 Flow Solver Adjoint Solver Gradient Calculation Update Airfoil Shape Modify Grid Repeat Process Until Convergence Flow Solver FLO03 Modified Runge Kutta Explicit Time Stepping. JST Scheme for Artificial Dissipation Local Time Stepping, Implicit Residual Smoothing, Multigrid Adjoint Solver The adjoint solver uses the same scheme as the flow solver Design Variables Surface Mesh Points Cost Function Inverse Design Drag Coefficient

9 Finite Volume Method A finite-volume methodology is used to discretize the integral form of the conservation laws. When using a discretization on a body conforming structured mesh, it is useful to consider a transformation to the computational coordinates (ξ,ξ ) defined by the metrics K ij = x i, J = det (K), K ξ j ij = ξ i x j. Figure illustrates the finite-volume mesh for cell (i, j). The values of the flow properties are stored at the cell centers marked by the red dot. Points between cell centers are marked as black dots and appear along cell boundaries and are identified by the ± to either i or j. The cell boundaries will also be referred to as flux faces.

10 Finite Volume Method (i, j ) ξ x ξ (i, j) (i, j) x (i, j ) Figure : Finite Volume Mesh for Cell (i)

11 Finite Volume Method 3 The Navier-Stokes equations can then be written in computational space as (Jw) t (F i F vi ) ξ i = 0 in D, () where the inviscid and viscous flux contributions are now defined with respect to the computational cell faces by F i = S ij f j and F vi = S ij f vj, and the quantity S ij = JKij represents the projection of the ξ i cell face along the x j axis. When equation () is formulated for each computational cell, a system of first order ordinary differential equations is obtained. Equation () can then be written for each computational cell as where (Jw) ij t R(w) i = 0, R(w) ij = F ξ F ξ F v ξ F v ξ. ()

12 Finite Volume Method 4 Then each partial derivative in equation () represents the net flux in each direction of the computational space. Each partial derivative of the convective and viscous flux gradients can then be represented in discrete form for each computational cell using a central second order discretization as F where the ± ξ = f i f i, (3) notation indicates that the quantity is calculated at the flux faces. Expand each partial derivative to its discrete form in equation () using the notation from equation (3) to yield R(w) ij = f i f i f i f i f vi f v i f v i f vi, (4) where f and f v are the convective and viscous fluxes.

13 Finite Volume Method 5 The scheme reduces to a second order accurate central difference scheme if the mesh was Cartesian. Schemes of this type generate oscillations around shock discontinuities and allow odd-even decoupling of the solution. To damp these high frequency errors, artificial dissipation terms are added to the convective and viscous fluxes. Equation (4) can then be rearranged to produce R(w) ij = h i h i h i h i, (5) where h denotes the numerical flux across the cell interfaces, h i = f i f v i d i, and d represents the artificial dissipation term. The values of the flow variables are stored at the cell centers, and can be regarded as cell averages. The convective flux f i at the cell face as shown in figure, is computed by taking the average of the flux contributions from each cell across the cell face as shown in the following equation f i = ( f i f ) i. (6)

14 Finite Volume Method 6 (i, j) (i, j) (i, j) Figure : Discretization of the Convective Fluxes

15 Finite Volume Method 7 On a Cartesian grid, the discretization produces a three-point second order central difference scheme for each flux gradient. Next we define the flux velocity in each cell as q = q = y ηi (ρu) i x η i (ρv) i ρ i y ηi (ρu) i x η i (ρv) i ρ i. Then the flux vectors can be formulated as ρ i q ρ i q f i = (ρu) i q y p ηi i (ρv) i q x p ηi i,and f i = (ρu) i q y p ηi i (ρv) i q x p ηi i. (ρe p) i q (ρe p) i q

16 Finite Volume Method 8 A numerical evaluation of the viscous flux requires an estimate of the partial derivative of velocity in the viscous stress tensor and the partial derivative of temperature from the Fourier Law of heat conduction. To evaluate the viscous flux at the cell face, we First compute the stress tensor and the heat flux components of the viscous flux at the end points (vertex) of the edge by employing a discrete Gauss theorem to the auxiliary control volume formed by the cell centers of the four cells containing the vertex (i, j ± ) as illustrated in figure 3. Second, the viscous flux across the cell face is computed by averaging the viscous fluxes at both ends of the edge. The viscous flux f vi at the cell face can be computed by taking the average of the fluxes computed at the cell vertex as shown below f vi = y η i f v i x ηi g v i, where f and g represent the fluxes at the mid-point of the cell face.

17 Finite Volume Method 9 (i, j ) (i, j ) (i, j) (i, j ) (i, j) (i, j ) (i, j) Figure 3: Auxiliary Control Volume for the Discretization of the Viscous Fluxes (i, j ) (i, j )

18 Finite Volume Method 0 The fluxes are computed by taking the average of the flux contributions from the cell vertex at (i, j ± ) as illustrated by the blue points in figure 3 and can be written as f v i = f vi f vi g v i = g vi g vi. Next we define the auxiliary control volume for the discretization of the viscous terms. The blue rectangle in figure 3 is formed by joining the cell centers of the four cells that share the cell vertex at (i, j ). The viscous fluxes at the cell vertex can then be expanded for two-dimensional flow as follows 0 0 f vi = τ xx τ xy uτ xx vτ xy k T x i g vi = τ yx τ yy uτ yx vτ yy k T y i.

19 Finite Volume Method The velocity components, u and v, the coefficient of thermal conductivity, k, and temperature T are averaged between the values from the four cells that form the auxiliary control volume. The following is an example of how any one of these parameters are computed u i = 4 (u i u i u i u i ). (7) The viscous stress tensors, τ xx and τ yy, act in the horizontal and vertical directions, normal to the faces of the auxiliary control volume. The shear stress, τ xy, is exerted in the xy plane. The method by which the viscous stress tensors and the temperature gradients are calculated require the following procedure. The steps will only be shown for the normal stress tensor, τ xx. The calculation for all other stress tensors will follow in a similar fashion.

20 Finite Volume Method First, expand the normal stress tensor as follows τ xxi = µ i u x i λ i u x v y i, (8) where the first and second coefficient of viscosities, µ and λ, are calculated by averaging the values between the four cells that form the auxiliary control volume as shown in equation (7). The first coefficient of viscosity, µ, is a combination of the laminar and turbulent viscosity coefficients defined as µ i = (µ lam µ turb ) i. The laminar coefficient of viscosity is calculated using the Sutherland equation and the eddy viscosity coefficient is calculated using the Baldwin-Lomax turbulence model. Then the velocity gradients are calculated by a transformation to the computational coordinates u x i = J i u ξ i y ηi u η i y ξi.

21 Finite Volume Method 3 Lastly, J is the volume of the auxiliary control volume and the velocity gradient in the computational domain, for an example, u, can be calculated by taking an average of the ξ velocity differences in the x direction between the top and bottom faces of the auxiliary control volume as shown below u ξ i = (u i u i ) (u i u i ). (9) The metric term y ηi, is calculated by taking the average of the difference between the y coordinate in the η direction from the front and back faces of the auxiliary control volume. The formulation described to compute the viscous flux, guaranties conservation and produces a second order accurate algorithm.

22 Finite Volume Method 4 In discrete form, the first order dissipation term can be written for the (i, j) face as where d () i = ν() Λ i (w i w i ), Λ i = ( λ ξi λ ξi ) and λ ξ and λ η are the scaled spectral radii of the flux Jacobian matrices in the computational domain. The purpose of multiplying the first order dissipative flux with the spectral radius is to control the magnitude of the dissipative term. In viscous flow, the flow field velocity in the ξ direction is approximately equal to the freestream velocity in the far-field region but approaches zero within the boundary layer. The magnitude of the dissipative term should be much smaller than the magnitude of the viscous flux gradient. Otherwise it would introduce an excessive amount of dissipation into the flow field and corrupt the quality of the solution.

23 Finite Volume Method 5 The ν () term also controls the amount of dissipation throughout the flow field and is made proportional to the normalized second difference of the pressure field. The term acts as a pressure gradient sensor and turns on at regions with large pressure changes such as shock waves. The sensor can be defined as ν () i = ɛ () max(σ i, σ i ), where σ i = p i p i p i p i p i p i. Unity is a typical value for ɛ (). The third derivative term can be expressed as where d (4) i = ν(4) Λ i (w i 3w i 3w i w i ), ν (4) = max [ 0, (ɛ (4) ν () ) ].

24 Finite Volume Method 6 The ν (4) sensor is defined to turn off in regions of high pressure gradient since it has the tendency to reintroduce oscillations around discontinuities. The ɛ (4) constant is defined such that the magnitude of ν () exceeds it around discontinuities. The sensor ν (4) then returns a value of zero around these regions and effectively turns off the third derivative term. A typical value for ɛ (4) is 3. The complete first and third order dissipative flux can be formulated as d i = d() i d(4) i.j d i = ν() Λ i (w i w i ) ν (4) Λ i (w i 3w i 3w i w i ). (0)

25 Optimal Design Suppose that performance is measured by a cost function I = B M (w, S) db ξ D P (w, S) dd ξ, () where db ξ and dd ξ are the surface and volume elements in the computational domain. Functions M and P depend on both the flow variables w and the metrics S defining the computational space. The design problem is now treated as a control problem where the boundary shape represents the control function, which is chosen to minimize I subject to the constraints defined by the flow equations. A shape change produces a variation in the flow solution δw and the metrics δs, which in turn produce a variation in the cost function with δi = B δm(w, S) db ξ D δp(w, S) dd ξ, () δm = M M δw w S δs, P P and δp = δw w S δs where the terms M P M and are computed with the metrics fixed, while w w S computed with the flow variables fixed. P and S are

26 Optimal Design The Navier-Stokes equations in computational space can be written as (Jw) t (F i F vi ) ξ i = 0 in D. A variation of the Navier-Stokes equations in steady-state can then be written as ξ i δ (F i F vi ) = 0. Here δf i and δf vi can also be split into contributions associated with δw and δs using the notation where, δf i = [F iw ] I δw δf iii, δf vi = [ F viw ]I δw δf v iii. [F iw ] I = S ij f j w, δf iii = δs ij f j.

27 Optimal Design 3 Multiplying by a co-state vector ψ, also known as a Lagrange Multiplier, and integrating over the domain produces D ψt ξ i δ (F i F vi ) = 0. If ψ is differentiable this may be integrated by parts to give B n iψ T δ (F i F vi ) db ξ D ψ T ξ i δ (F i F vi ) dd ξ = 0. Since the left-hand expression equals zero, it may be subtracted from the variation of the cost function () to give δi = B [ δm ni ψ T δ (F i F vi ) ] db ξ D δp ψt δ (F i F vi ) dd ξ. (3) ξ i

28 Optimal Design 4 Now, since ψ is an arbitrary differentiable function, it may be chosen in such a way that δi no longer depends explicitly on the variation of the state vector δw. The gradient of the cost function can then be evaluated directly from the metric variations without having to re-compute the variation δw resulting from the perturbation of each design variable. The variation δw may be eliminated from (3) by equating all field terms that are multiplied by δw to produce a differential adjoint system governing ψ: ψ T ξ i [ Fiw F viw ] I P w = 0 in D. (4) The corresponding adjoint boundary condition is produced by equating all boundary terms that are multiplied by δw in equation (3) to produce n i ψ T [ F iw F viw ]I = M w on B. (5)

29 Optimal Design 5 The remaining terms from equation (3) then yield a simplified expression for the variation of the cost function that defines the gradient δi = B D δp II ψt ξ i [δf i δf vi ] dd II ξ. (6) { δmii n i ψ T [δf i δf vi ] II } dbξ The details of the formula for the gradient depend on the way in which the boundary shape is parameterized as a function of the design variables, and the way in which the mesh is deformed as the boundary is modified. Using the relationship between the mesh deformation and the surface modification, the field integral is reduced to a surface integral by integrating along the coordinate lines emanating from the surface. Thus the expression for δi is finally reduced to δi = B GδF db ξ, where F represents the design variables and G is the gradient, which is a function defined over the boundary surface.

30 Optimal Design 6 The boundary conditions satisfied by the flow equations restrict the form of the left-hand side of the adjoint boundary condition (5). The boundary contribution to the cost function M cannot be specified arbitrarily. It must be chosen from the class of functions that allow cancellation of all terms containing δw in the boundary integral of equation (3). On the other hand, there is no such restriction on the specification of the field contribution to the cost function P, since these terms can always be absorbed into the adjoint field equation (4) as source terms.

31 Continuous Adjoint The weak form of the Euler equations for steady flow is D φ T ξ k F k dd = B n kφ T F k db, (7) where the test vector φ is an arbitrary differentiable function and n k is the outward normal at the boundary. If a differentiable solution w is obtained to this equation, then it can be integrated by parts to give D φt F k ξ k dd = 0. Suppose now that we wish to control the surface pressure by varying the airfoil shape. For this purpose, it is convenient to retain a fixed computational domain. Variations in the shape then result in corresponding variations in the mapping derivatives defined by K.

32 Continuous Adjoint Introduce the cost function I = B W (p p d ) ds, where p d is the desired pressure. The design problem is now treated as a control problem where the control function is the airfoil shape, which is chosen to minimize I subject to the constrains defined by the flow equations. A variation in the shape causes a variation δp in the pressure and consequently a variation in the cost function δi = B W (p p d ) δp ds B W (p p d ) δds. Since p depends on w through the equation of state, the variation δp is determined from the variation δw. Define the Jacobian matrices as A k = f k w, C k = S kl A l. (8)

33 Continuous Adjoint 3 The weak form of the equation for δw in the steady state becomes where D φ T ξ k δf k dd = B (n kφ T δf k )db, δf k = C k δw δs kl f l, which should hold for any differentiable test function φ. This equation may be added to the variation in the cost function, which may now be written as δi = B W (p p d ) δp ds D ψ T B W (p p d ) ξ k δf k dd B (n kψ T δf k )db. δds

34 Continuous Adjoint 4 On the airfoil surface B W, n = 0. Therefore, δf = 0 S δp S δp 0 0 δs p δs p 0. Since the weak equation for δw should hold for an arbitrary choice of the test vector φ, we are free to choose φ to simplify the resulting expressions. Therefore we set φ = ψ, where the costate vector ψ is the solution of the adjoint equation ψ t CT k ψ ξ k = 0 in D. (9) At the outer boundary, incoming characteristics for ψ correspond to outgoing characteristics for δw. Consequently we can choose boundary conditions for ψ such that n k ψ T C k δw = 0.

35 Continuous Adjoint 5 If the coordinate transformation is such that δs is negligible in the far-field, then the only remaining boundary term is B W ψ T δf dξ. Thus, by letting ψ satisfy the boundary condition, ψ j n j = p p d on B W, (0) where n j are the components of the surface normal n j = we find finally that S j Sj S j, δi = B W (p p d ) δds D ψ T ξ k δs kl f l dd B W (δs ψ δs ψ 3 ) p dξ.

36 Continuous Adjoint 6 The convective adjoint flux is discretized using a second order central spatial discretization. The first step is to expand equation (9) for a two-dimensional problem ψ t CT ψ ξ C T ψ ξ = 0. Define ξ = ξ and η = ξ. Then the continuous adjoint fluxes can be discretized as V ψ i t = [ C T i (ψ i ψ i ) C T ] i (ψ i ψ i ) d i d i d i d i, where V is the cell area and d i has the same form as equation (0).

37 Continuous Adjoint 7 From equation (8), the Jacobian fluxes can be expanded as where C T i = y ηi A T i x ηi A T i and C T i = y ξi A T i x ξi A T i, y ηi = y ηi y η i, x ηi = x ηi x η i, A T i = f w T, A T i = g w T. In order to reduce the number of subscripts and simplify the notation, the Euler Jacobian matrices are defined as follows A T i = C T i, B T i = C T i, A T i = AT i, B T i = AT i. () Finally, the convective continuous adjoint flux can be written as R(ψ) = [ÂT i (ψ i ψ i ) B T i (ψ i ψ i ) ]. ()

38 Continuous Adjoint 8 In computational coordinates, the viscous terms in the Navier Stokes equations have the form F vi ξ i = ξ i ( Sij f v j ). Computing the variation δw resulting from a shape modification of the boundary, introducing a Lagrange vector ψ and integrating by parts produces B ψt ( δs j f v j S j δf v j) dbξ D ψ T ξ i ( δsij f v j S ij δf v j ) ddξ, where the shape modification is restricted to the coordinate surface ξ = 0 so that n = 0, and n =. The viscous terms will be derived under the assumption that the viscosity and heat conduction coefficients µ and k are essentially independent of the flow, and that their variations may be neglected.

39 Continuous Adjoint 9 The derivation of the viscous adjoint terms is simplified by transforming to the primitive variables w T = (ρ, u, u, p) T, because the viscous stresses depend on the velocity derivatives u i x j, while the heat flux can be expressed as where κ = k R = γµ P r(γ ). κ x i The relationship between the conservative and primitive variations is defined by the expressions p ρ δw = Mδ w, δ w = M δw which make use of the transformation matrices M = w w and M = w w.

40 Continuous Adjoint 0 These matrices are provided in transposed form for future convenience M T = u u u i u i 0 ρ 0 ρu 0 0 ρ ρu γ M T = (γ )u i u i u ρ u ρ 0 0 (γ )u ρ 0 0 (γ )u ρ γ. The conservative and primitive adjoint operators L and L corresponding to the variations δw and δ w are then related by with D δwt Lψ dd ξ = D δ wt Lψ dd ξ, L = M T L, so that after determining the primitive adjoint operator by direct evaluation of the viscous portion of, the conservative operator may be obtained by the transformation L = M T L.

41 Continuous Adjoint In order to make use of the summation convention, it is convenient to set ψ j = φ j for j =,. Then the contribution from the momentum equations is B φ k (δs j σ kj S j δσ kj ) db ξ D φ k ξ i (δs ij σ kj S ij δσ kj ) dd ξ. () The velocity derivatives in the viscous stresses can be expressed as with corresponding variations u i = u i ξ l = S lj x j ξ l x j J u i ξ l δ u i x j = S lj J I ξ l δu i u i ξ l II δ S lj J.

42 Continuous Adjoint The variations in the stresses are then δσ kj = { µ { µ [ Slj J [ δ δu ξ k S lk δu l J ξ j l ( Slj J ) uk ξ l δ ( S lk J ] ) uj ξ l λ ] λ [ S δ lm jk J [ ξ l δu m δ jk δ ( S lm J ]} I ) um ξ l ]} II. As before, only those terms with subscript I, which contain variations of the flow variables, need be considered further in deriving the adjoint operator. The field contributions that contain δu i in equation () appear as D φ k ξ i S ij µ S lj J δu k S lk ξ l J S lm δu j λδ jk ξ l J δu m ξ dd ξ. l

43 Continuous Adjoint 3 This may be integrated by parts to yield D δu k ξ l δu D m ξ l µ S lj S ij J φ k ξ i λδ jk S lm S ij J dd ξ D δu j φ k ξ i dd ξ, ξ l µ S lk S ij J φ k ξ i dd ξ where the boundary integral has been eliminated by noting that δu i = 0 on the solid boundary. By exchanging indices, the field integrals may be combined to produce D δu k ξ l S lj µ S ij J φ k ξ i S ik J φ j ξ i S im λδ jk J φ m ξ dd ξ, i which is further simplified by transforming the inner derivatives back to Cartesian coordinates D δu k ξ l S lj µ φ k φ j x j x k φ m λδ jk x dd ξ. () m

44 Continuous Adjoint 4 The boundary contributions that contain δu i in equation () may be simplified using the fact that on the boundary B so that they become δu i = 0 if l = ξ l B φ ks j µ S j J δu k S k ξ J S m δu j λδ jk ξ J δu m ξ db ξ. (3) Together () and (3) comprise the field and boundary contributions of the momentum equations to the viscous adjoint operator in primitive variables.

45 Continuous Adjoint 5 In order to derive the contribution of the energy equation to the viscous adjoint terms it is convenient to set ψ 4 = θ, Q j = u i σ ij κ x j p ρ, where the temperature has been written in terms of pressure and density. The contribution from the energy equation can then be written as B θ (δs jq j S j δq j ) db ξ D θ ξ i (δs ij Q j S ij δq j ) dd ξ. (4) The field contributions that contain δu i,δp, and δρ in equation (4) appear as D θ S ij δq j dd ξ = θ S ξ D ij {δu k σ kj u k δσ kj i ξ i κ S lj δp J ξ l ρ p δρ ρ ρ dd ξ. (5)

46 Continuous Adjoint 6 The term involving δσ kj may be integrated by parts to produce D δu k ξ l S lj µ θ θ u k u j x j x k θ λδ jk u m x m dd ξ, where the conditions u i = δu i = 0 are used to eliminate the boundary integral on B. Notice that the other term in (5) that involves δu k need not be integrated by parts and is merely carried on as D δu kσ kj S ij θ ξ i dd ξ. (6) The terms in expression (5) that involve δp and δρ may also be integrated by parts to produce both a field and a boundary integral.

47 Continuous Adjoint 7 The field integral becomes D δp ρ p ρ δρ ρ ξ l κ S lj S ij J θ dd ξ ξ i which may be simplified by transforming the inner derivative to Cartesian coordinates D δp ρ p ρ δρ ρ ξ l S lj κ θ dd ξ. (7) x j The boundary integral becomes κ B δp δρ ρ p ρ ρ S j S ij J θ ξ i db ξ. This can be simplified by transforming the inner derivative to Cartesian coordinates κ B δp δρ ρ p ρ ρ S j J θ x j db ξ,

48 Continuous Adjoint 8 and identifying the normal derivative at the wall and the variation in temperature n = S j, (8) x j δt = R to produce the boundary contribution δp ρ p ρ δρ ρ, B kδt θ n db ξ. (9) This term vanishes if T is constant on the wall but persists if the wall is adiabatic.

49 Continuous Adjoint 9 There is also a boundary contribution left over from the first integration by parts (4) which has the form since u i = 0. B θδ (S jq j ) db ξ,,where Q j = k T x j, (0) Notice that for future convenience in discussing the adjoint boundary conditions resulting from the energy equation, both the δw and δs terms corresponding to subscript classes I and II are considered simultaneously. If the wall is adiabatic so that using (8), T n = 0, δ (S j Q j ) = 0, and both the δw and δs boundary contributions vanish.

50 Continuous Adjoint 0 On the other hand, if T is constant T ξ l = 0 for l =, so that Q j = k T x j = k Thus, the boundary integral (0) becomes S l j J T ξ l = k S j J T ξ. B kθ S j J ξ δt δ S j J T ξ db ξ. () Therefore, for constant T, the first term corresponding to variations in the flow field contributes to the adjoint boundary operator and the second set of terms corresponding to metric variations contribute to the cost function gradient.

51 Continuous Adjoint Collecting together the contributions from the momentum and energy equations, the viscous adjoint operator in primitive variables can be expressed as ( Lψ) = p ρ ξ l ( Lψ) i = ξ S lj l ξ S lj l θ σ ij S lj ( Lψ) 4 = ρ ξ l S lj κ θ x j µ µ φ i φ j x j x i θ θ u i u j x j x i ξ l for i =, S lj κ θ x j φ k λδ ij x k θ λδ ij u k x k The conservative viscous adjoint operator may now be obtained by the transformation. L = M T L.

52 Continuous Adjoint In the case of the continuous adjoint boundary condition, equation (0) constrains the values of the normal adjoint velocities. The tangential adjoint velocity, ψ, and ψ 4 do not appear; therefore, assigning a zero value for these variables does not violate equation (0). This results, however, in poor convergence for the adjoint equation because it is an over-specification of the adjoint boundary condition. A satisfactory boundary condition may be formulated as follows: ψ i, = ψ i, ψ i, = ψ i, n ( (p pd ) n ψ i, n ψ 3i, ) ψ 3i, = ψ 3i, n ( (p pd ) ψ i, n ψ 3i, ) ψ 4i, = ψ 4i,, (3) where n i = S i Sj. The subscripts (i, ) and (i, ) in the above equations denote S j cells below and above the wall. Here, the first and fourth costate variables below the wall are set equal to the corresponding values above the wall and the tangential adjoint velocities above and below the wall are equated.

53 Continuous Adjoint 3 If the drag is to be minimized, then the cost function is the drag coefficient, I = C d = c y C p BW ξ dξ cos α c x C p BW ξ dξ sin α. A variation in the shape causes a variation δp in the pressure and consequently a variation in the cost function, δi = c c B W C p B W C p y x cos α ξ ξ sin α δpdξ y x δ cos α δ sin α dξ. (4) ξ ξ As in the inverse design case, the first term is a function of the state vector, and therefore is incorporated into the boundary condition, where the integrand replaces the pressure difference term in equation (3). The second term is added to the gradient term.

54 Discrete Adjoint The discrete adjoint equation is obtained by applying control theory directly to the set of discrete field equations, following the same sequence of steps. The resulting equations depend on the details of the scheme used to solve the flow equations. To formulate the discrete adjoint equation, we first take a variation of the residual term. From equation (5), the first variation can be written as with δr(w) ij = δh i δh i δh i δh i, (5) δh i = δf i δf v i δd i δh i = δg i δg vi δd i, (6) where f and g are the convective flux gradients, f v and g v are the viscous flux gradients, and d is the artificial dissipation term.

55 Discrete Adjoint Next, we pre-multiply the variation of the discrete residual by the Lagrange Multiplier and sum the product over the computational domain to produce the following nx i= ny j= ψ T i δr(w) i. (7) Thirdly, similarly to the the primary steps taken to produce the continuous adjoint equation, equation (7) is added to the variation of the discrete cost function, δi = δi c nx i= ny j= ψ T i δr(w) i, where δi c is the discrete cost function, and R(w) i is the residual term. To develop the discrete adjoint equations, the discrete counterpart to the integration by parts, summation by parts is required. To produce the final set of discrete adjoint equations, expand the δr(w) term for cell (i, j) and the adjacent four cells. Then multiply the variation of the residual by the Lagrange multiplier, ψ i. Lastly, collect any term that is multiplied by δw i.

56 Discrete Adjoint 3 A full discretization of the equation would involve discretizing every term that is a function of the state vector. The development cost of the method grows rapidly with the order and size of the stencil of the discretization scheme. The numerical scheme we employ to solve the flow field equations utilize a central second order spatial discretization to evaluate the flux gradients in each direction. The fluxes are averaged at the flux faces before the flux gradients are computed. This is equivalent to a three-point stencil to evaluate the flux gradient in each direction. First, we will only consider the contribution from δf i to equation (6) and ultimately to equation (5). From equation (6), the first variation of the convective flux computed at the flux face can be written as δf i = ( δf i δf ) i. (8)

57 Discrete Adjoint 4 Now expand δf i, δf i = δ y f ηi i x g ηi i = y ηi δf i δy ηi f i x ηi δg i δx ηi g i = y ηi = f w δw i δy ηi f i x ηi g w δw i δx ηi g i y A ηi i x B ηi i δw i δy f ηi i δx g ηi i, (9) where A and B are the convective flux Jacobians f can be expanded to produce the following w and g w. Similarly, the δf i term δf i = y A ηi i x B ηi i δw i δy f ηi i δx g ηi i. (30) Note here that the metric terms in equation (9) and (30) are identical since the plus and minus fluxes are evaluated along the (i, j) edge. The only difference between the two equations are the state vector terms.

58 Discrete Adjoint 5 Substituting of equations (9) and (30) into equation (8), keeping only terms that are multiplied to the variation of the state vector and neglecting the variation of the dissipative and viscous fluxes for now, equation (6) can be expanded to δh i = y ηi A i x ηi B i y ηi A i x ηi B i δw i δw i. (3) In the η direction, δg i can be written as δg i = δg i δg i = δ y ξi f i x ξi g i = = y ξi A i x ξi B i y ξi A i x ξi B i ( δg i δg i ), where δw i δy ξi f i δx ξi g i, δw i δy ξi f i δx ξi g i.

59 Discrete Adjoint 6 Then δh i can be expressed as δh i = y ξi A i x ξi B i y ξi A i x ξi B i δw i δw i. (3) We now have all the necessary terms to formulate the variation of the convective flux δr(w) i. Substitution of equation (3) and (3) into equation (5) will produce δr(w) i = δh i δh i δh i δh i = y A ηi i x B ηi i δw i y A ηi i x B ηi i y A ηi i x B ηi i δw i y A ηi i x B ηi i y ξi A i x ξi B i y ξi A i x ξi B i δw i δw i δw i δw i y ξi A i x ξi B i y ξi A i x ξi B i δw i δw i.

60 Discrete Adjoint 7 Simplify the above equation and reorder the terms to produce an equation for the contribution of the convective flux from the field equations to the variation of the residual in cell (i, j), δr(w) i = y ηi A i x ηi B i y ηi A i x ηi B i δw i δw i y ηi y η i y ξ i y ξi A i x x ηi η x i ξ i x ξi y ξi A i x ξi B i y ξi A i x ξi B i δw i δw i B i δw i. (33) Note that the equation above has contributions from all four adjacent cells. In order to simplify the notation, δr(w) i will be represented by δr i.

61 Discrete Adjoint 8 The variation of the residual vector from the adjacent cells such as δr i, δr i, etc., have contributions from the (i, j) cell. For example, if equation (33) is written for cell (i, j) and only the δw i terms are shown, then the variation of the residual vector for cell (i, j) can be written as δr(w) i = y ηi A i x ηi B i δw i The next step is to pre-multiply the variation of the residual vector by the transpose of the Lagrange multiplier vector and sum the product over the entire domain. This step leads to the following equation nx i= ny j= ψ T i δr i = ψ T i δr i ψ T i δr i ψ T i δr i ψ T i δr i ψ T i δr i. (34)

62 Discrete Adjoint 9 The discrete domain spans from i = nx and j = ny, where nx and ny are the maximum cell points. Next we substitute the expansions for the variation of the residual terms from equation (33) for each term in equation (34) and collect the δw i terms to produce nx i= ny j= ψ T i δr i = ψ T i ψ T i ψ T i ψ T i ψ T i y ηi A i x ηi B i y ηi A i x ηi B i y ηi y η i y ξ i y ξi x x ηi η x i ξ i x ξi y ξi A i x ξi B i A i B i y ξi A i x ξi B i δw i (35)

63 Discrete Adjoint 0 Reordering the terms in equation (35) leads to the following equation nx i= ny j= ψ T i δr i = ( ψ T i ψt i ) y ηi A i x ηi B i ( ψ T i ψt i ) y ηi A i x ηi B i ( ψ T i ψt i ) y ξ i ( ψ T i i ) ψt y ξ i A i x ξi B i A i x ξi B i δw i

64 Discrete Adjoint Take a transpose of the equation and the adjoint convective flux can then be written as R (ψ) = y ηi AT i x η i BT i (ψ i ψ i ) y ηi AT i x η i BT i (ψ i ψ i ) y ξi A T i x ξ i B T i (ψ i ψ i ) y ξi A T i x ξ i B T i (ψ i ψ i ). Next, define the flux Jacobian matrices for the total flux across the cell face in the computational domain as A T i = y η i AT i x η i BT i and B T i = y ξ i AT i x ξ i BT i.

65 Discrete Adjoint Finally, the discrete convective flux can be represented by the following expression [ÂT R (ψ) = i (ψ i ψ i ) ÂT i (ψ i ψ i ) B T i (ψ i ψ i ) B T i (ψ i ψ i ) ]. (36) Note here that if an average of the metrics evaluated at either flux faces were used in the definition of the flux Jacobian matrices for the total flux across the wall in the computational domain, then ÂT i would reduce to ÂT i. Equation (36) would reduce to the following R (ψ) = [ÂT i (ψ i ψ i ) B T i (ψ i ψ i ) ]. (37) Equation (37) is identical to the discretization of the continuous convective flux gradient defined in equation (). This illustrates that the discretization of the continuous and discrete convective fluxes are similar and only differ in the manner the metrics are calculated in each cell.

66 Discrete Adjoint 3 From equation (37), in the limit that the mesh width reduces to zero, the discrete adjoint convective flux can be written as lim ξ 0, η 0 R (ψ) = [ÂT i (ψ i ψ i ) B T i (ψ i ψ i ) ]. The second order central difference of the Lagrange Multipliers can then be reduced as lim ξ 0 ψ i ψ i = ψ ξ. Finally, the discrete adjoint convective flux term can be written in continuous form as R (ψ) = lim ξ 0, η 0 ÂT ψ ξ R (ψ) = ÂT ψ ξ B T ψ η = CT ψ B T ψ η. ξ C T ψ ξ = C T k ψ ξ k The expression above is identical to the continuous adjoint equation defined in eqn (9).

67 Discrete Adjoint 4 The derivation of the discrete viscous adjoint fluxes are simlar to the discrete convective adjoint fluxes illustrated in the previous section; however, the task of producing the viscous counterpart is challenging due to the additional terms in the Navier-Stokes equations. Since only the viscous fluxes will be considered in this section, the first variation of the total residual for the control volume (i, j) can be simplified to the following equation δr(w) ij = δf vi δf v i δf v i δf vi, where the total flux through the (i, j) flux face can be defined as δh i = δf v. i Next the variation of the viscous flux at the cell face can be computed by taking an average of the fluxes at the cell vertex and shown below δf = δ vi y f ηi v δ i x ηi g v, (38) i where f v i = f vi f vi, g v i = g vi g vi.

68 Discrete Adjoint 5 By the chain rule, equation (38) can be expanded to produce terms that are multiplied to the variation of the state vector and shape function. Such an expansion would produce the following equation δf vi = δy η i f v i y ηi δf v i δx ηi g v i x ηi δg v i. (39) We will choose to ignore the metric variations for the rest of the section and concentrate only on expressions that produce terms that are multiplied by the variation of the state vector. The variation of the viscous flux contribution from the cell vertex (i, j ) can be defined as 0 δf vi = δτ xx δτ xy δuτ xx uδτ xx δvτ xy vδτ xy δk T x kδ T x i. (40)

69 Discrete Adjoint 6 The variation of the viscous fluxes at the other cell vertexes are defined in a similar fashion. Due to the large number of terms that needs to be considered in the derivation of the discrete viscous adjoint fluxes, the contributions from the momentum and energy equations will be considered in separate sub-sections. We will concentrate our efforts on the contributions from the momentum equation. First, rewrite equation (39) without the variation due to metric terms and substitute f v and g v terms with the average of the viscous fluxes at the cell vertexes, δf vi = y η i δ f vi f vi x η δ i g vi g vi. Next rearrange the terms to produce the following equation δf vi = y ηi δf v i y ηi δf v i x ηi δg v i x ηi δg v i = [ ] δf i δf i.

70 Discrete Adjoint 7 We now concentrate on the momentum equation in the ξ direction. Then δf can be expressed as δf i = y δf ηi v i x δg ηi v i = y δτ ηi xx i x δτ ηi yx i. (4) The next step is the variation of the stress tensor terms. The purpose of this exercise is to illustrate the procedure and not to show the full derivation; therefore, the expansion will only be carried out for the τ xx term. The viscosity coefficients will be treated as constants in the derivation and therefore its variations are zero and will be neglected in the following derivation, δτ xxi = µ i δ u x i λ i δ u x i δ v y i.

71 Discrete Adjoint 8 Note The stress tensor terms are functions of the primitive variables, ρ, u, v, T, and not the state vector, w, which is comprised of ρ, ρu, ρv, ρe. In the expansion of the variation of the stress tensor terms, we seek ultimately to produce an equation that is a function of the variation of the primitive variables, δu, δv,.... It will be shown that once the discrete viscous adjoint fluxes are formed, it will be transformed back to a form that is multiplied not by the variation of the primitive variables but by the variation of the state vector. This will allow us to sum the discrete viscous adjoint fluxes to the discrete convective and artificial dissipation fluxes. To simplify the procedure we will attempt to complete the derivation by only collecting terms that are multiplied by the variation of the velocity in the x direction, δu.

72 Discrete Adjoint 9 First, substitute the velocity gradient terms into the above expression for the variation of the stress tensor to produce δτ xxi = µ λ J i [ (y η y ξ ) i δu i (y η y ξ ) i ( y η y ξ ) i δu i (y η y ξ ) i δu i ], δu i where J is the cell volume. The variation of the shear stress term, δτ xy can be derived in a similar manner. The variation of the flux at the cell vertex (i, j ) from equation (4) can be expressed as δf i = y ηi x ηi µ λ J i [(y η y ξ ) δu i (y η y ξ ) δu i (y η y ξ ) δu i ( y η y ξ ) δu i ] µ [(x ξ x η ) δu i (x ξ x η ) δu i J i (x ξ x η ) δu i ( x ξ x η ) δu i ]. (4)

73 Discrete Adjoint 0 All metrics terms are evaluated at the (i, j ) vertex. In equation (4), the first two lines are contributions from the variation of the normal stress term, δτ xx, and the third and fourth lines are contributions from the variation of the shear stress term, δτ yx. The expression for δf i can be produced by subtracting one from the j subscript from equation (4). Thus, the total flux across the (i, j) flux face can be formulated using the following expression δh i = δf v i = [ ] δf i δf i. Finally, the variation of the total residual in cell (i, j) can be expressed as δr(w) i = δf vi δf v i δf v i δf vi = [ ] δf i δf i [ ] δg i δg i [ ] δf i δf i [ ] δg i δg i

74 Discrete Adjoint Here δg is defined as δg i = y ξi δf vi x ξi δg vi = y ξi δτ xxi x ξi δτ yxi. The next step is to pre-multiply the variation of the residual by the transpose of the Lagrange Multiplier, ψ T, and sum the product over the computational domain to produce nx i= ny j= ψ T i δr i = ψ T i δr i ψ T i δr i ψ T i δr i ψ T i δr i ψ T i δr i ψ T i δr i ψ T i δr i ψ T i δr i ψ T i δr i. (43) The total contribution towards the residual from the viscous fluxes requires information from all eight cells that surrounds cell (i, j). Terms that are multiplied by δu i are collected to produce the total discrete viscous adjoint residual for the second adjoint equation. This is due to the fact that only the δu i terms are being considered and not the complete viscous flux.

75 Discrete Adjoint After some lengthy algebra the discrete adjoint stress tensor can be expressed as ϖ xxi = µ i ψ x i λ i ψ x ψ 3 y i. Note here the remarkable similarity between the Navier-Stokes equation viscous stress tensor expressed in equation (8) and discrete viscous adjoint stress tensor shown above. The velocities, u and v, are simply replaced by the second and third adjoint variables. The adjoint variable gradients are expressed as ψ x i = J i y ξi y ηi ψ η ψ ξ i y ηi i y ξi ψ η ψ ξ i i. (44)

76 Discrete Adjoint 3 We now focus on the contributions from the energy equation. From equation (40) the contribution from the energy equation can be expressed as δf vi = δuτ xx uδτ xx δvτ xy vδτ xy δk T x kδ T x i. (45) The contribution can be divided into three parts: variation of the stress tensors (uδτ xx vδτ xy ) i variation of the velocities (δuτ xx δvτ xy ) i contribution from the heat addition terms. ( δk T x ) T kδ x i.

77 Discrete Adjoint 4 First, consider the contribution of the variation of the stress tensor terms from the energy equation. The variation of the flux at the cell vertex can be written as δf i = y ηi δf v i uδτ xxi x ηi uδτ yxi = y ηi x ηi δg v i vδτ xyi vδτ yyi. (46) Equation (46) is similar to equation (4) of the previous section. The main difference between the two equations is the fact that the variation of the stress tensor terms are multiplied by the velocity, u and v. Thus the derivation of the contribution of these terms to the discrete viscous adjoint fluxes follows the derivation of the discrete adjoint stress tensor from the previous section. The contribution of these terms to the discrete viscous adjoint flux can be expressed as ϑ xxi = (uµ) i ψ 4 x i λ i u ψ 4 x v ψ 4 y i

78 Discrete Adjoint 5 Second, consider the contribution of the variation of the velocity components to the discrete viscous adjoint fluxes. The first step is to express the variation of the flux at the cell vertex with only contributions from terms multiplied by the variation of the velocity components. From equation (45), the variation of the flux at the cell vertex can be expressed as δf i = y δf ηi v i x δg ηi v i = y ηi δu i τ xx i δv i τ xy i x ηi δu i τ yx i δv i τ yy i. (47) The flow field velocities are calculated at the cell vertex by averaging the values of the velocities from the four cells that share the same vertex.

79 Discrete Adjoint 6 Concentrating our efforts on the variation of the u velocity component and replacing the equation for the velocity at the cell vertex, equation (47) can be simplified to δf i = y δu ηi i τ xx i x δu ηi i τ yx i = y ηi 4 τ xx i x τ ηi yx i [δu i δu i δu i δu i ]. Since these terms are contributions from the energy equation, they would only be multiplied by the last component of the vector representing the transpose of the Lagrange Multiplier in equation (43). After a series of algebraic manipulations, similar to the procedure used from the previous section, the second contribution from the energy equation to the discrete viscous adjoint equation can be written as ϱ xi = 4 τ xxi ψ 4 x i τ yxi ψ 4 y i. The Lagrange Multiplier gradients are defined by equation (44).

80 Discrete Adjoint 7 The last contribution from the energy equation to the discrete viscous adjoint flux is from the variation of the heat addition term. From equation (45) the variation of the viscous flux can be expressed as δf vi = δk T x kδ T x i. If the coefficient of thermal conductivity is treated as a constant, then the only remaining term is the variation of the temperature gradient. Represent temperature as one of the primitive variables, (ρ, u, v, p) to produce δf vi = kδ T x = γ i k x = γ ρ δp p ρ δρ kδ x p ρ i. i

81 Discrete Adjoint 8 After similar algebraic manipulations, the third contribution from the energy equation to the discrete viscous adjoint equation can be written as ε xi = k i ψ 4 x i.

82 Discrete Adjoint 9 Collecting together the contributions from the momentum and energy equations, the viscous discrete adjoint operator in primitive variables for two-dimensional flow can be expressed as ( Lψ) = p ρ (y η x η ) ε x p ρ (y η x η ) ε y p ρ (x ξ y ξ ) ε y i i p ρ (x ξ y ξ ) ε x i i ( Lψ) = [y η (ϖ xx ϑ xx ) x η (ϖ xy ϑ xy )] i [x ξ (ϖ xy ϑ xy ) y ξ (ϖ xx ϑ xx )] i ϱ xi ϱ xi ϱ yi ϱ yi [y η (ϖ xx ϑ xx ) x η (ϖ xy ϑ xy )] i [x ξ (ϖ xy ϑ xy ) y ξ (ϖ xx ϑ xx )] i ( Lψ) 3 = [y η (ϖ yx ϑ yx ) x η (ϖ yy ϑ yy )] i [x ξ (ϖ yy ϑ yy ) y ξ (ϖ yx ϑ yx )] i [y η (ϖ yx ϑ yx ) x η (ϖ yy ϑ yy )] i [x ξ (ϖ yy ϑ yy ) y ξ (ϖ yx ϑ yx )] i ( Lψ) 4 = ϱ yi ϱ yi ρ (y η x η ) ε x ϱ xi ϱ xi ρ (y η x η ) ε y ρ (x ξ y ξ ) ε y i i ρ (x ξ y ξ ) ε x i i

83 Discrete Adjoint 30 The conservative viscous adjoint operator may now be obtained by the transformation L = M T L. The transformation matrices M and M T are provided below. M T = u u u i u i 0 ρ 0 ρu 0 0 ρ ρu γ M T = (γ )u i u i u ρ u ρ 0 0 (γ )u ρ 0 0 (γ )u ρ γ.

84 Discrete Adjoint 3 A complete variation of the fluxes would require a variation of every term that is a function of the state vector. Thus a variation of the sensor terms, ν () and ν (4), requires a variation of the second difference of the pressure field and a variation of the spectral radius would then require a variation of the velocity and the speed of sound term which in itself is a function of the pressure and density fields. This would require an extensive amount of work, and since the magnitude of the dissipative terms is lower than the convective and viscous fluxes, the sensor terms and the spectral radii can be treated as constants. Accordingly a variation of the artificial dissipation term would result in the following equation, δh i = δd i δh i = ν() Λ i (δw i δw i ) ν (4) Λ i (δw i 3δw i 3δw i δw i ).

85 Discrete Adjoint 3 A complete variation of the fluxes would require a variation of every term that is a function of the state vector. Thus a variation of the sensor terms, ν () and ν (4), requires a variation of the second difference of the pressure field and a variation of the spectral radius would then require a variation of the velocity and the speed of sound term which in itself is a function of the pressure and density fields. This would require an extensive amount of work, and since the magnitude of the dissipative terms is lower than the convective and viscous fluxes, the sensor terms and the spectral radii can be treated as constants. Accordingly a variation of the artificial dissipation term would result in the following equation, δh i = δd i δh i = ν() Λ i (δw i δw i ) ν (4) Λ i (δw i 3δw i 3δw i δw i ).

86 Discrete Adjoint 33 Next, we examine the variation of the total residual in the control volume. Since we desire only to formulate the dissipation in the ξ direction, then the variation of the total residual in the control volume can be represented as δr(w) ij = δh i δh i = ν () Λ i (δw i δw i ) ν (4) Λ i (δw i 3δw i 3δw i δw i ) ν () Λ i (δw i δw i ) ν (4) Λ i (δw i 3δw i 3δw i δw i ). (48) We must then pre-multiply the variation of the residual by the transpose of the Lagrange Multiplier, sum the product over the computational domain, and isolate terms that are multiplied by the variation of state vector, δw i, in the (i, j) control volume. Since the blended first and third order dissipation scheme used in this work requires a five point stencil, then it is necessary to include the variation of the residual from these five cells.

87 Discrete Adjoint 34 Thus the equation can be represented as nx i= ny j= ψ T i δr i = ψ T i δr i ψ T i δr i ψ T i δr i ψ T i δr i ψ T i δr i. (49) Now substitute the variation of the residual terms from equation (48) into equation (49). Only terms that are multiplied by the variation of the state vector in the (i, j) cell, δw i, are shown. nx i= ny j= ψ T i δr i = ψ T i ν(4) i 3.jΛ i 3 ψ T i ψ T i ψ T i [ ν () ] i.jλ i 3ν(4) i.jλ i ν(4) i 3.jΛ i 3 [ ν () i.jλ i 3ν(4) i.jλ i ν() i.jλ i [ ν (4) i 3.jΛ i 3 ν() i.jλ i 3ν(4) i.jλ i 3ν(4) i.jλ i ] ] ψ T i ν(4) i 3.jΛ i 3.

88 Discrete Adjoint 35 Then two sets of terms, one for each flux face of the control volume can be formed as follows nx i= ny j= ψ T i δr i = [ ν () ( i.jλ i ψ T i ) ( ψt (4) i ν i 3.jΛ i 3 ψ T i ) ψt i ν (4) ( i.jλ i ψ T i ψt i [ ν () i.jλ i ( ψ T i ψt i ν (4) i.jλ i ( ψ T i ψt i. ) ν (4) ) ν (4) ( i.jλ i ψ T i ψt i ( i.jλ i ψ T i ψt i ) ) ] ) ν (4) i 3.jΛ i 3 ( ψ T i ψt i ) ] We can now define the discrete adjoint blended first and third order artificial dissipation scheme as D i = ( ν() i.jλ i ψ T i ) ψt (4) i ν ν (4) i.jλ i ( ψ T i ψt i ( i 3.jΛ i 3 ψ T i ψt i ) ν (4) i.jλ i ( ψ T i ψt i ) ). (50)