Analytical solutions for compartmental models of contaminant transport in enclosed. spaces

Transcription

1 ESGI8 Analytical solutions for compartmental models of contaminant transport in enclosed Problem presented by Simon Parker DSTL Executie Summary Understanding the transport of hazardous airborne materials within buildings and other enclosed is important for predicting and mitigating the impacts of deliberate terrorist releases of chemical and biological materials. Multizone models proide an approach to modelling the contamination leels in enclosed but in certain cases they can be computationally expensie. Alternatie methods are being explored at DSTL that inole the direct solution of the contaminant dynamics equation. The Study Group was asked to identify the limits to this alternatie approach and to explore its extension. Version 1.1 Noember 3, 211 iii+24 pages i

2 ESGI8 Report author Tristram Armour (Industrial Mathematics KTN) Contributors Tristram Armour (Industrial Mathematics KTN) Elif Başkaya (Karadeniz Technical Uniersity) Stephen Cowley (Uniersity of Cambridge) Ishak Cumhur (Rize Uniersity) Patricio Farrell (Uniersity of Oxford) Dariusz Grala (CIAMSE & Uniersity of Warsaw) Jens Graesen (Technical Uniersity of Denmark) Siân Jenkins (Uniersity of Bath) James Kirkpatrick (Uniersity of Oxford) Karin Mora (Uniersity of Bath) Margarita Schlackow (Uniersity of Oxford) Süleyman Sengül (Karadeniz Technical Uniersity) Xiaoting Wang (Uniersity of Cambridge) Dae Wood (Uniersity of Warwick) ESGI8 was jointly organised by The Uniersity of Cardiff The Knowledge Transfer Network for Industrial Mathematics and was supported by The Engineering and Physical Sciences Research Council ii

3 ESGI8 Contents 1 Introduction Background Multizone models Analytical solutions Questions Work Diagonalisation Alternatie methods of calculating the exponential of a matrix Total Exposure Inerse problem - quadratic minimisation Inerse problem: Control theory method Transient growth Conclusions Multizone model with lags Final comments A Results from special cases 19 A.1 Alternatie model for finding the toxic load A.2 Duct model B Tools and codes 23 B.1 EigTool B.2 Schur-Parlett Bibliography 24 iii

4 ESGI8 1 Introduction 1.1 Background (1.1.1) Understanding the transport of hazardous airborne materials within buildings and other enclosed is important for predicting and mitigating the impacts of deliberate terrorist releases of chemical and biological materials. Because such materials may be acutely toxic or infectious it is important to understand how concentrations may change with time to understand the hazards that different scenarios may pose. It is also releant to the study of accidental releases of industrial materials and the impact of enironmental pollutants on indoor air quality. (1.1.2) A range of different numerical modelling approaches are regularly used to study these problems as well as experimental methods. Computational fluid dynamics (CFD) can be used for detailed studies of air and contaminant moement within enclosed. Howeer, CFD methods require highly detailed input data and hae significant model creation and execution times. This can make them impractical for whole building studies in some cases. (1.1.3) Multizone models (CONTAM, COMIS) proide an alternatie approach where the building is diided into a series of well-mixed olumes connected by paths through which air and contaminants can pass. These models hae the adantage that they are quicker to execute than CFD models and typically require less input information. The contaminants are normally considered dilute in such approaches. Typical model size is of the order of 1-1 zones, although 1 or more may be required in some cases. (1.1.4) Multizone models sole the air flow through the network (typically using a non-linear pressure soler) for a series of quasi-steady states. The contaminant dispersion resulting from the air flow solution and the contaminant initial and boundary conditions is then calculated. Whilst these models are well deeloped and hae been alidated for a range of studies, they rely on numerical methods which can become time consuming for large studies (e.g. Monte Carlo analysis). In addition, little insight is gained into the system behaiour using a numerical approach. 1.2 Multizone models (1.2.1) The transport of a dilute contaminant between a number of zones of fixed olumes can be considered as an example of a compartmental system and we describe it by a system of linear ordinary differential equations, ẋ = Ax + Bu (1) 1

5 ESGI8 where x is the ector of the contaminant concentrations (mass / olume) in each zone of the system, A is the matrix of interzone and exhaust flows normalised by zone olumes (defined below) and Bu describes the mapping of external concentrations and internal source terms onto the system. In the general case, B = V 1 and u = Q ext x ext + u int, where Q ext is a matrix of flow rates into the building, x ext is the ector of external contaminant concentrations at the entry points into the building and u int u int (t) is the source term of the contaminant within the building. (1.2.2) A is defined as follows: A = V 1 Q (2) where V is a diagonal matrix where V i,i is the olume of zone i in [m 3 ] and V i,j = for i j. Q 1,1 Q 1,2 Q 1,n. Q Q = 2,1 Q 2, Qn 1,n Q n,1 Q n,n 1 Q n,n where Q i,j is the flow into zone i from zone j when i j and Q j,j = ( n i=1 Q i,j Q,j ) is the flow out of zone j, where Q,j is the flow of air out of the system from zone j. Note that i and j take alues from 1 to n and the index represents the exterior to the building. All flow rates hae units of [m 3 s 1 ]. Figure 1 shows the general case of a multizone system with labelled olumes and flow rates. (1.2.3) Note that since A is diagonally dominant, we will hae negatie or zero eigenalues in all systems. (1.2.4) A complete description is gien in reference [1]. In reality there will be no control oer the external concentration or internal source terms. Howeer, it may be possible to control the olumetric flow into the building. Since this is incorporated within the term u some control oer the input may be achieed, although it is likely that this would also change A. One related application is the case where an airborne decontaminant is introduced into the building to remediate contaminated building surfaces. In that case one problem of interest would be how to control u to achiee a certain x. (1.2.5) In standard multizone models this system of equations is integrated numerically to gie concentrations based on the initial conditions. For a single scenario such a calculation is typically fast to carry out. Howeer for some applications, such as the optimisation of detector placement or (3) 2

6 ESGI8 Q 1, Q,1 Q 2,1 V 1 V 2 Q 1,2 Q 2,3 Q,2 Q 2, Q 1,i Q 1,3 Q 3,1 Q 2,i Q i,1 Q i,2 Q 3,2 Q 3, Q i,3 Q,i Q,3 V 3 Q 3,i V i Q i, Figure 1: Ventilated and connected multizone system the interpretation of sensor data, many thousands of indiidual simulations may need to be carried out. For large numbers of these repeated cases A will be constant, with only ariation in the initial concentration or source terms u. In some cases there may be large differences in zone olumes, or secondary effects which introduce a wide range of timescales and require small timesteps oer long periods to sole the stiff equations. 1.3 Analytical solutions (1.3.1) The solution to (1) can be written as follows x(t) = e A(t) x() + t e A(t µ) B(µ)u(µ)dµ (4) where t is the current time and µ is a ariable of integration. (1.3.2) In the case where the state transition matrix A is diagonalisable the exponential term can be expressed as e At = Se Λt S 1 (5) where S is the matrix of eigenectors and Λ is a matrix with the eigenalues arranged on the diagonal. (1.3.3) This explicit solution for the contaminant concentrations as well as the exposure (the integral of the concentration x for any zone) in the form of a sum of exponentials is particularly useful as it allows us to calculate the concentration at any future time (for a constant A) without iteration. When screening a large number of scenarios this direct calculation can sae time. 3

7 ESGI8 (1.3.4) Furthermore, in the special case where u is constant in time, we hae the following analytical solution. x(t) = Se Λ(t τ) S 1 x(τ) SΛ 1 (I e Λ(t τ) )S 1 Bu (6) (1.3.5) The eigenectors and eigenalues proide insight into the behaiour of the system such as the late phase decay rate and concentration ratio. The eigenalue of the smallest magnitude controls the final decay rate of the system. The dependence of this eigenalue on system properties is of interest. Complex eigenalues appear to arise from recirculating flow paths and result in damped oscillations. (1.3.6) Howeer, it was recognised by DSTL that A may not always be diagonalisable. For example, a collection of zones with identical olumes in series and with the same flow through each of them leads to a non-diagonalisable case. This is an important case, since a series of such zones can be used in the CONTAM multizone software to represent a length of duct work to improe the time resolution of the contaminant transport. 1.4 Questions (1.4.1) These are the questions posed to the Study Group by DSTL: (a) Are there other cases where A is not diagonalisable? Is it possible to characterise the types of systems that are diagonalisable and nondiagonalisable? (b) For the diagonalisable case is it possible to bound the alues of the smallest magnitude eigenalue based on the properties of the system matrix A or related system properties such as the total exhaust flow and olume? We hae seen examples where this alue is both larger and smaller than the system flushing rate (the air change rate). (c) For the general case where A is not diagonalisable it seems that it is possible to construct an analytical solution to state transition matrix e At using the Jordan canonical form. Is it possible to derie a useful analytical form for the concentration solution for any matrix A? Many authors warn against the use of the Jordan form for practical calculations - does this rule out this approach for the solution for any A? For the simpler cases is there a physical interpretation? (d) Is it possible to sole the inerse problem? In other words, if we hae measurements of concentration (x i ) in one or more zones, can we establish any information about the source terms or external concentrations u? (e) What are the practical limits to using solutions of the form (5) for large systems? We hae encountered numerical problems in calculating the concentration at short times when compared to iteratie 4

8 ESGI8 schemes, for systems with around 3 zones for large condition numbers. Are there alternatie approaches that can aoid these problems? (f) We hae explored some results for a nested two-zone case where it can be shown that the exposure in each zone is the same as the external exposure. Can this be extended to more zones? Are there other systems which lead to special case solutions? (g) Is it possible to write down solutions for cases when the input function u is not constant (e.g. periodic, linear ramp etc.)? (h) Can we calculate the total exposure, x(t)dt? (i) Can we calculate the toxic load, x p (t)dt where p > 1? (j) Can we find the maximum of x(t)? 2 Work 2.1 Diagonalisation (2.1.1) From a linear maths point of iew, there are ways of finding out if a matrix A is diagonalisable or not. The minimal polynomial of A, m(x), is defined as the polynomial of least degree that satisfies m(a) =. It can be shown that A is diagonalisable if and only if m(x) is decomposable into distinct linear factors. To find the minimal polynomial of A, you start with the characteristic polynomial, χ(x) = det(a xi) which satisfies by definition, χ(a) =. The minimal polynomial is either the characteristic polynomial or another polynomial that includes some of the factors from the characteristic polynomial. (2.1.2) What sort of models in the form lead to the case when A is not diagonalisable? It was recognised that A is not diagonalisable if we hae zone layouts that introduce wae-like elements to the solution of x. Indeed, if the building contains a long air duct, or if there is a chain of identical rooms with the same air flow leads to a situation where A has repeated eigenalues and in these cases we would hae to find an alternatie method to find e At. Appendix A.2 works through the solution to a single duct case. (2.1.3) A further complication can arise when we model fast flowing air such as may be found in an air handling system. The small olume with a high flow rate compared to the large olumes and low flow rates inside the building will create a system with large and smaller eigenalues which will lead to stiff systems for large multizone models. To treat the specific case of the time lag in air handling systems an extension to the multizone model described in Section 3.1 may be worth deeloping. 5

9 ESGI8 2.2 Alternatie methods of calculating the exponential of a matrix (2.2.1) The paper by Moler and Van Loan [2] gies a ery good summary of methods for calculating the exponential of a matrix. They suggest methods which are likely to be most efficient for problems inoling large matrices and repeated ealuation of e At. The method inestigated by the Study Group inoles the Schur decomposition and the Cayley-Hamilton Theorem. (2.2.2) The Schur decomposition turns a square matrix into an upper triangular matrix. Further transformations (laid out in [5]) turn the matrix into a block triangular structure. T T A = S S 1, (7)... T m where T i is triangular with eigenalues close to each other and S is well conditioned. (2.2.3) If the triangular blocks T i are small, we may use the Caley-Hamilton theorem. Consider a k k matrix A and the characteristic polynomial for ta: We can write t (s) = det(ta si). (8) exp(s) = t (s) Q t (s) + R t (s), (9) where Q t is an analytic function and the reminder R t is a polynomial of degree k 1, k 1 R t (s) = r l (t) s l. (1) (2.2.4) Caley-Hamilton s theorem that states that t (ta) =, implies that l= k 1 exp(ta) = t (ta) Q t (ta) + R t (ta) = R t (ta) = r l (t)t l A l. (11) (2.2.5) We want to determine the coefficients r l (t), especially in the case of nearly defectie eigenalues. We consider the case of a triangular matrix l= T = λi + E, (12) 6

10 ESGI8 where E is triangular with small alues in the diagonal. Since λi and E commute we hae exp(tt) = exp(tλi) exp(te) = exp(tλ) exp(te), (13) which is enough to determine exp(te). (2.2.6) First we do the simplest possible case, namely a 2 2 matrix: T = ( ) λ + ɛ c = λ ɛ (2.2.7) As t (t(λ ± ɛ)) =, we hae ( ) ( ) λ ɛ c + λ ɛ and E = ( ) ɛ c ɛ (14) (2.2.8) This gies us the linear equations (2.2.9) That is, with the solution exp( tɛ) = R t ( tɛ) = r (t) r 1 (t)tɛ, (15) exp(tɛ) = R t (tɛ) = r (t) + r 1 (t)tɛ. (16) ( ) ( ) 1 tɛ r (t) 1 tɛ r 1 (t) ( ) r (t) = 1 r 1 (t) 2tɛ = ( tɛ tɛ 1 1 ( ) exp( tɛ) exp(tɛ) ) ( exp( tɛ) exp(tɛ) (17) ). (18) r (t) = r 1 (t) = tɛ exp( tɛ) + tɛ exp(tɛ) = cosh(tɛ), 2tɛ (19) exp(tɛ) exp( tɛ) = sinh(tɛ). 2tɛ tɛ (2) Thus exp(te) = cosh(tɛ)i + sinh(tɛ) te (21) tɛ and, using that T = λi + E ( exp(tt) = exp(tλ) cosh(tɛ)i + sinh(tɛ) ) te. (22) tɛ 7

11 ESGI8 (2.2.1) When ɛ we obtain ( ) 1 tc lim exp(tt) = exp(tλ) (I + te) = exp(tλ) = exp(lim t T). (23) ɛ 1 ɛ (2.2.11) Here we hae a 3 3 example matrix λ + 2ɛ c 12 c 13 λ 2ɛ c 12 c 13 T = λ ɛ + δ c 23 = λ + ɛ + δ c 23. λ ɛ δ λ ɛ δ (24) (2.2.12) We now hae the linear equations 1 t(λ + 2ɛ) t 2 (λ + 2ɛ) 2 r (t) exp(2tɛ) 1 t(λ ɛ + δ) t 2 (λ ɛ + δ) 2 r 1 (t) = exp(tλ) exp( t(ɛ δ)). 1 t(λ ɛ δ) t 2 (λ ɛ δ) 2 r 2 (t) exp( t(ɛ + δ)) (25) 2.3 Total Exposure (2.3.1) Recall our linear system, where dx dt = A x + B u(t), (26) A = V 1 Q, B = V 1, u = Q ext x ext + u int. (27) When there is a single external concentration that impacts on the building x ext is a scalar and Q ext becomes a column ector which satisfies the following: (2.3.2) We can see that Q 1. + Q ext =. (28) 1 and A 1 B = Q 1 (29) Q 1 Q ext = 1.. (3) 1 8

12 ESGI8 (2.3.3) The solution with initial condition x() = x is and we hae t x(t) = e At x + e At e Aτ B u(τ) dτ (31) e At dt = A 1 e At. (32) (2.3.4) As all the eigenalues of A are strictly negatie, lim t e ta = implies that t e At dt = A 1 e At. (33) τ 1 t 111 dτ dt τ dt dτ τ t Figure 2: The two integration orders t Therefore x(t) dt = e At x dt + t e At e Aτ B u(τ) dτ dt. We switch the integration order in the double integral, see Figure 2, and get ( ) = A 1 x + e At dt e Aτ B u(τ) dτ = A 1 x + = A 1 x A 1 B = A 1 x τ A 1 e Aτ e Aτ B u(τ) dτ u(τ) dτ u int1 (τ) x ext (τ) dτ Q 1. dτ. u intn (τ) (34) 9

13 ESGI8 2.4 Inerse problem - quadratic minimisation (2.4.1) Consider the equations presented in Section 2.3. (2.4.2) We now assume we hae a series of obserations/measurements y i and want to determine a load u(t) that explains the obserations. Obseration i is the concentration in room k i at time t i, i.e., y i = x ki (t i ). So we need to calculate x(t i ) in order to check our guess u(t). We consider the case where x = and u = φ j (t)e l, e l is the lth standard basis ector, and φ j (t) = { 1 t [(j 1), j ], t / [(j 1), j ]. (2.4.3) We can now perform the integration in (4) and obtain (35) x(i ) =, i < j, (36) x(i ) = (e A I)A 1 B e l, i = j, (37) x(i ) = e A(i j) (e A I)A 1 B e l, i > k. (38) (2.4.4) We next consider the case u = k j=1 jφ j (t)e l and obtain C x( ) 11 e l C 12 e l... C 1k e l C 22 e l... C 2k e l 1. = (39) x(k )... C kk e k l where C ij is a n n matrix C ij = e A(i j) (e A I)A 1 B. (4) (2.4.5) Obsere that C ij e l is simply the lth column in C ij and that we can obtain the matrices C 1j by recursion C 1,j+1 = e A C 1j. (41) (2.4.6) It is worth noting that the eigenalues for A all lie in the left half plane so the exponential e A has norm less than 1. That in turn implies that we do not amplify errors by the multiplication in (41). Let Π be the projection on to the obsered rooms, i.e., Π is obtained from the unit matrix I by remoing the rows corresponding to the rooms without sensors. We then 1

14 ESGI8 hae the following equation to determine the load u = k j=1 jφ j (t)e l. y 1 y =. y m = ΠC 11 e l ΠC 12 e l... ΠC 1k e l ΠC 22 e l... ΠC 2k e l ΠC kk e k l = C l. (42) (2.4.7) Obsere that we can obtain C l from the large matrix C 11 C C 1k C C 2k C kk (43) by deleting some rows and columns. (2.4.8) The least square solution to (42) is and the error is l = (C T l C l ) 1 C T l y, (44) ɛ l = y C l l. (45) (2.4.9) We hae an error ɛ l foreach l =,..., n and the l which gies the smallest error is our prediction for where the contaminant originates. (2.4.1) We consider the situation where we for each time step i hae a set of measurements y i = Πx(i ). We can now sole the least square problems (42) for l =,..., n in each time step and thereby get both a prediction for origin of the contaminant, l, and for the leel of the contaminent, l. (2.4.11) We can only hae positie terms in u(t) so we should sole the following constrained optimisation problem. such that minimise i y C l l 2 (46) i, i = 1,..., k. (47) This is a standard problem, a quadratic functional with linear constraints, and is not much harder to sole than the unconstrained ersion. 11

15 ESGI8 2.5 Inerse problem: Control theory method (2.5.1) This is an alternatie method to soling the inerse problem using a control theory result and data assimilation. Gien x(t ) at time T >, can we find u(t)? The controllability Gramian is gien by: Q T = T e Aτ BB e A τ dτ (48) (2.5.2) Gien x(t ), T > and x(t δ ), if detq T, then we can steer x(t δ ) to x(t ) in time T using the control u(t) = B e A (T t) Q 1 T (x(t ) eat x()) t [, T ] (49) and Q T = T ea t B Be At dt is defined as the Controllability Gramian and = T [8]. The control, (49) is not necessarily unique in its ability to steer x to x 1 in time T >. It minimises the energy u(t) dt, oer all possible u(t) that steer x(t) from x to x 1 in time T >. Identifying ũ(t) would proide a possible control for determining the future progression of the contaminant throughout the building and also allow the identification of an interior or an exterior source for the contaminant. (2.5.3) In order for the control ũ(t) to be identified, both x and x 1 need to hae contaminant readings for eery room in the building. Howeer, due to practical constraints, detectors may not be placed in eery room in the building. This means that x and x 1 are incomplete data sets. This means that the missing entries need to be estimated in some way to allow ũ(t) to be identified. Data assimilation is a process which could be used to estimate the missing alues. (2.5.4) Data assimilation requires a model for the physical process and a set of obserations of the true physical system. The aim of the process is to find the least squares solution between the obserations and the results of the model, gien some initial estimate for the initial conditions. The result of the process proides an initial condition for the model which, when run, will produce a simulation of the model which will minimise the square error between the obserations and the preious results of the model created using the estimated initial conditions. (2.5.5) Let time t = denote the time the contaminant is first detected and let time T > be another time in the future where contaminant leels are also known. Also, assume detq T and that there are n N rooms in the building. The model for this problem is gien by (4), where the control is gien by (49), with x = x() R n and x 1 = x(t ) R n. The model will proide complete sets of data to allow the identification of ũ(t) R n. 12

16 ESGI8 (2.5.6) Let y : R R m, t y(t) denote an obseration of the contaminant leels in the building at time t, where the ith element represents the concentration of the contaminant in the ith room at time t. This means that there are m N rooms with detectors in. Let y i R m denote the ith obseration of the contaminant leels in the building, where y = y() and y N = y(t ). Also, denote x i R n as the corresponding state of the model at the time of the ith obseration. This means that x = x() and x N = x(t ). (2.5.7) In order to generate the model states for comparison with the obserations, estimates for x() and x(t ) need to be made in order to simulate ũ(t), so x(t) can be simulated. Let x b R m denote an estimate for x() and x e R m denote an estimate for x(t ). These estimates can be made based upon the obserations at these points in time, y() and y(t ) respectiely. (2.5.8) As the obserations y i are incomplete sets of data, in order to allow for the comparison of y i with x i, the matrix H i R m n is required to select only the elements of x i that are present in y i. This means that H i is linear and due to the assumption that each detector remains in the same position for each reading, H i = H i where H is constant. In control theory, H would normally be denoted by C. (2.5.9) Data assimilation also requires knowledge on the accuracy of the obserations, the initial state estimate, x b R n and the final state estimate, x e R n. The accuracy of the ith obseration is represented by the ith coariance matrix R i R m m. As the detectors used to take each obseration remain the same oer time, R i = R i, a constant coariance matrix. G R n n (usually denoted by B) and E R n n represent the coariance matrices for x b and x e respectiely. The j, kth element of these matrices is determined by the coariance of the jth and kth elements of the ectors they correspond to. For example, (R) j,k = co((y i ) j, (y i ) k i. It is assumed though that each coariance matrix is a diagonal matrix by assuming that the cross-coariances are always zero. For example (R) j,k = co((y i ) j, (y i ) k ) = for j k. More information on the definitions of the different ariables and the ideas and deriation behind data assimilation can be found in [9]. (2.5.1) The least squares solution for x b and x e are achieed through data assimilation by minimising the following cost functions with respect to x R n and z R n respectiely J 1 (x) = (x x b ) T G 1 (x x b ) + J 2 (z) = (z x e ) T E 1 (z x e ) + 13 N (y i Hx i ) T R 1 (y i Hx i ), (5) i= N (y i Hz i ) T R 1 (y N+1 i Hz i ), (51) i=

17 ESGI8 where z i = x N+1 i. (2.5.11) These both need to be minimised simultaneously to find the best solutions. (5) and (51) are minimised when J 1 (x) = and J 2 (z) =, where in this case J 1 (x) = 2G 1 (x x b ) + 2 J 2 (z) = 2E 1 (z x e ) + 2 N R 1 (y i Hx i ), (52) i= N R 1 (y N+1 i Hz i ). (53) (2.5.12) Data assimilation is usually used to find an improed estimate for the initial condition for the model of the system which best fits the obserations of the true system. This is what minimising (5) aims to achiee for x(). Howeer, we wish to also find a better estimate for x(t ) as well. Due to the control function ũ(t), x(t) is steered from the current estimate for x() to the current estimate for x(t ). This means that any results from the model start and end with these estimates respectiely. This means that we can iew the results in reerse order as results from the model where the current estimate for x(t ) was the initial condition. This way, the same form of data assimilation can be applied to find an improed estimate for x(t ) as for finding an improed estimate for x(). (2.5.13) The algorithm for minimising (5) and (51) would begin by initially choosing x = x b for minimising (5) and z = x e for minimising (51). The Newton method is a possible method which could be used to identify x and z that satisfy (52) and (53) respectiely. For example, denote the current best estimate for x() by x (k) and the next best estimate by x (k+1) and the current best guess for x(t ) by z (k) and the next best estimate by z (k+1). Then (2.5.14) Algorithm i= x (k+1) = x (k) [ J 1 (x (k) )] x J 1 (x (k) ), (54) z (k+1) = z (k) [ J 2 (z (k) )] z J 2 (z (k) ). (55) (a) Estimate x() and x(t ) based on obserations y() and y(t ) respectiely and denote these x b and x e respectiely. (b) Set x (k) = x b and z (k) = x e for k =. (c) Calculate ũ(t) steering from x (k) to z (k) in time T >, using (49). Check that detq T. (d) Calculate x(t) using (4) at the time of each obseration y i using ũ(t) found in the preious step. (e) Minimise J 1 (x (k) ) to find x (k+1). 14

18 ESGI8 (f) Calculate J 1 (x (k+1) ) and J 2 (z (k) ) to see if they e been minimised. If not, continue. Otherwise, stop. (g) Calculate ũ(t) steering from x (k+1) to z (k) in time T >, using (49). Check that detq T. (h) Calculate x(t) at the time of each obseration y i using ũ(t) found in the preious step. (i) Minimise J 2 (z (k) ) to find z (k+1). (j) Calculate J 1 (x (k+1) ) and J 2 (z (k+1) ) to see if they e been minimised. If not, continue. Otherwise stop. (k) Add 1 to k and repeat steps 3 to 1 until J 1 (x) and J 2 (z) are minimised for some x and z. This would find a better estimate for x b and x e. These can then be used to create ũ(t). One disadantage to this method is that there is no constraint on making each concentration of contaminant in the system positie. There is also a need to estimate coariance matrices and the initial states for the system. (2.5.15) The second part of the inerse problem inoled identifying whether the contaminant originated from an interior or an exterior source. The control function was proposed to be constructed from the following function u(t) = Q ext x ext (t) + u int (t). (56) The equation (4), together with (49), can be written using (56) as t x(t) = e At x + e At e Aτ B[Q ext x ext (τ) + u int (τ)]dτ, where t t = e At y + e At e Aτ BQ ext x ext (τ)dτ + e At z + e At e Aτ Bu int (τ)dτ, = y(t) + z(t), t y(t) = e At y + e At e Aτ BQ ext x ext (τ)dτ, (57) t z(t) = e At z + e At e Aτ Bu int (τ)dτ. (58) (2.5.16) The proportion of each element of x(t) assigned to each element of y(t) and z(t) would be fixed oer time and be based on the probability of the contaminant in the room haing an interior or an exterior source. Room i has a probability of α i 1 that any contaminant which began entering the building in this compartment was from an external source and a probability of 1 α i of coming from an internal source. Hence (y) i (t) = α i (x) i (t), (z) i (t) = (1 α i )(x) i (t). 15

19 ESGI8 For example, suppose room j has no exterior flows, then α j =. (2.5.17) If detq T, then possible control functions to steer (57) from y() to y(t ) and to steer (58) from z() to z(t ) in time T > are gien by x ext (t) = (Q ext B) e A (T t) Q 1 T (y(t ) eat y()), (59) ũ int (t) = B e A (T t) Q 1 T (z(t ) eat z()), (6) such that ũ(t) = Q ext x ext (t) + ũ int (t) where ũ(t) is a possible control that steers (4) from x() to x(t ). 2.6 Transient growth (2.6.1) We can find upper and lower bounds for the maximum of x(t) for a gien u. For some time t, the solution for t > T, (2.6.2) Define the following T x(t) = e At x() + e At e Aµ B(µ)u(µ)dµ (61) = e At y. (62) Ax A = sup x x (63) and x(t) G(t) = sup x() x() = eat. (64) (2.6.3) Suppose A is diagonalisable, with eigenalues λ 1,..., λ n where λ 1 is the eigenalue with the smallest magnitude and λ n is the eigenalue with the largest magnitude. Note that Re(λ 1 ) is the spectral abscissa α(a) since all eigenalues are negatie. Then F s.t. A = F ΛF 1, where Λ =diag(λ 1, λ 2,..., λ n ). Let j be the eigenector associated with the j-th eigenalue. Then Note that e At j = e λ jt j. (65) e At j j = e λ jt = e Re(λ j)t (66) and so we hae the following lower bound, G(t) e α(a)t, [7]. 16

20 ESGI8 (2.6.4) For the upper bound [7] e At = e F ΛF 1t = F e Λt F 1 (67) = F F 1 e Λt (68) F F 1 e α(a)t. (69) (2.6.5) When A is not diagonalisable, we can look at pseudospectra to find lower bounds for G max = max t G(t). Let ɛ >, define R(z) = (zi A) 1 and C ɛ = {z : R 1 }. (7) ɛ Then define β(z) = {max Re(z) : z C ɛ }. Hille-Yosida theorem states that, 3 Conclusions G max sup ɛ> 3.1 Multizone model with lags β(ɛ) ɛ. (71) (3.1.1) A possible multizone model extension is to deal with the disparate timescales introduced by air handling systems by using delay differential equations. Take the following example zone layout: air duct x 1 x 2 x n Figure 3: Multizone model with lag 17

21 ESGI8 (3.1.2) Suppose we say that the air takes τ seconds to get from zone 1 to zone n. We hae the following set of delay differential equations dx 1 dt = Q 1x Q 1 x 1 + Q 21 x 1 αx 1 (t) (72) dx j dt = Q j,j 1x j 1 Q j x j + Q j+1,j x j (73) dx n = Q n,n 1 x n 1 Q n x n + αx n (t τ n ). dt (74) (3.1.3) Taking all lags in the system into account, the linear system has the following form dx dt = Ax(t) + j dx dt = Ax(t) j B j x(t τ j ) (75) dx B j τ j (t) +... (76) dt If the delays are small (with respect to eacuation time of the building) we find that dx dt = Ax(t) j dx B j τ j (t) (77) dt dx dt = (I + B jτ j ) 1 Ax (78) dx dt = A x. (79) (3.1.4) Note that this is equialent to the original problem except that A is not diagonally dominant. 3.2 Final comments (3.2.1) The Study Group went some way to answer many of the questions posed by DSTL. We hae identified the types of systems when A is not diagonalisable. For other cases when DSTL were haing trouble with certain large systems, the disparate timescales in the system led to a stiff system. The Study Group suggested two methods to soling these kinds of problems. One way would be to use a suitable preconditioner (or equialently, a suitable matrix decomposition) for A. The other, is to reformulate the problem using delay differential equations. 18

22 ESGI8 (3.2.2) The inerse problem was tackled in two different ways. One is based on a result from control theory and data assimilation, the other, leads to a single minimisation problem of a quadratic functional. Further testing is required to see which method would be more suitable for DSTL. (3.2.3) A neat result was obtained related to the total exposure, x(t)dt, when the contaminant s source is from outside the building. Under the model assumptions, the total exposure is the same in all zones. (3.2.4) An alternatie model was deeloped that could lead to a way of calculating the toxic load, x p (t)dt for p > 1. This is described briefly in Appendix A.1. (3.2.5) Finally, a method of finding the lower bound for the transient growth of the system was described in Section 2.6. A upper bound and a lower bound can be obtained when A is diagonalisable. A Results from special cases A.1 Alternatie model for finding the toxic load (A.1.1) Consider a steady state flow which carries some contaminant by pure adection, i.e., no diffusion. The concentration outside is a function of time only, c = f(t), and it is carried through the area of interest along the stream lines, see Figure 4. c(x, t) f(t) Figure 4: The streamlines of the flow through the area of interest. (A.1.2) We assume we know the flow and denote the elocity by u(x). Furthermore, we assume we hae an incompressible flow, so u =. Then the concentration c(x, t) of the contaminant is gien by the equation c t + u c =. (8) (A.1.3) We now consider a point x on one of the streamlines and let s denote arclength along the streamline. Assuming the concentration is zero at time t = and for t we hae 19

23 ESGI8 d ds c(x, t) dt = d c(x, t) dt = ds c(x, t) dt = c t (x, t) dt =. (81) (A.1.4) So the total exposure is constant along the streamlines and is in particular equal to the total exposure at a point outside that area of interest. That is, c(x, t) dt = f(t) dt. (82) (A.1.5) The moments c(x, t) m satisfies the same equation (c m ) t + u (c m ) =. (83) Thus c(x, t) m dt = f(t) m dt. (84) (A.1.6) The time integrated aerage in some area Ω satisfies 1 Vol(Ω) Ω c(x, t) m dv dt = 1 = Vol(Ω) 1 Vol(Ω) Ω Ω f(t) m dt dv = c(x, t) m dt dv f(t) m dt. (85) (A.1.7) So the aerage of the moments is constant, but the moments of the aerage is in general not constant ( ) m 1 c(x, t) dv dt Vol(Ω) Ω f(t) m dt. (86) A.2 Duct model (A.2.1) We consider a model, where we hae n rooms. The air is flowing in one direction with the same flow rate q between each pair of compartments. The olume in each room is constant. That is, our matrices are of the following form 2

24 ESGI V = q... q q... Q = q q. (A.2.2) The matrix A = V 1 Q is not diagonalisable. Howeer, using one modified Jordan block, we can easily analytically compute the solution of the equation t x(t) = e At x() + e At e Aτ B(τ)u(τ)dτ, where in our case we hae x() =, B = V 1 and u = qx e 1. Here e 1 is the first canonical basis ector and x is the concentration of the inflowing contaminant from outside into the first room. (A.2.3) In order to compute the concentration x(t), we need to look at the matrix exponential e At. We find Similarly, we hae e At = e qt 1... qt ( qt )n 1 (n 1)! ( qt )n 2 (n 2)!... qt 1. e At = e qt So that we can write, 1... qt ( qt )n 1 (n 1)! ( qt )n 2 (n 2)!... qt 1. x(t) = qx eat 21 t e Aτ e 1 dτ.

25 ESGI8 (A.2.4) Substituting µ = q τ, the integral reduces to q x(t) = x e At t = x e At ( ( ) e µ 1, µ, µ2 2!,..., µ n 1 T dµ (n 1)! ),..., 1 Γ(n, qt ) ) T dµ, 1 Γ(1, qt (n 1)! where Γ(m, qt ) for m = 1,..., n denotes the incomplete Gamma function. Note that for an integer m, we hae Γ(m, qt m 1 qt ) = (m 1)! e k= ( qt )k k! (A.2.5) Using the eigenalue form for e At we can then write the contaminantconcentration in the last room, i.e. the nth component of x(t) as x n (t) = x n m=1 ( ( q m 1 t)n m ( q t)k (n m)! k! k=. e qt This function starts at and as time progresses it tends to the alue of x, which is what we expect to see. ) (A.2.6) Note that in a general case when A is not diagonalisable, i.e. its minimal polynomial is not expressible as a product of distinct linear factors, you can write it as a matrix-product P JP 1, where J is in Jordan block form, i.e. it consists of Jordan blocks on the diagonal and P is the transition matrix. Each Jordan block is a matrix with one eigenalue on the diagonal, and ones on the superdiagonal. The size of the blocks corresponds to the multiplicity of the eigenalues. Each exponentiation of the Jordan matrix has the adantage that the operations stay within the Jordan blocks. Then an explicit expression for x(t) can be obtained similarly to the diagonalisable case, after expressing the e Jt in terms of the eigenalues and a polynomial matrix, similarly as we hae done for e At with the only difference, that the matrix is now upper triangular. The ealuation is then analogous to the diagonalisable case, but we integrate oer the product of an exponential function and a polynomial (as done aboe). 22

26 ESGI8 Concentration in the last room x n for n = 1, 3, 4, 1 rooms as well as the expected concentration x n. B Tools and codes B.1 EigTool (B.1.1) Pseudosprectra software -aailable at: B.2 Schur-Parlett (B.2.1) Nick Higham s MATLAB code for the Schur-Parlett method: 23

27 ESGI8 Bibliography [1] State-space methods for calculating concentration dynamics in multizone buildings, S Parker, V Bowman. Building and Enironment (211). [2] Nineteen dubious ways to compute the exponential of a matrix, C Moler, C Van Loan. Society for Industrial and Applied Mathematics (23). [3] Transient growth and why we should care about it R Bale, R Goindarajan. Resonance (May 21). [4] Computing the matrix exponential - The Cayley-Hamilton Method D Rowell. Lecture notes (24) [5] A Schur-Parlett algorithm for computing matrix functions P Daies, N Higham. MIMS (23) [6] Spectra and pseudospectra: The behaiour of nonnormal matrices and operators L Trefethen, M Embree. Bulletin of the American Mathematical Society (27) [7] Psuedospectra of linear operators L Trefethen. SIAM Reiew (1997) [8] MA346 Lecture Notes, Uniersity of Bath[online] H Logemann, 28. [9] Data Assimilation Concepts and Methods[online] European Centre for Medium- Range Weather Forecasts. Reading. F Bouttier, P Courtier, Aailable from: lecture_notes/pdf_files/assim/ass_cons.pdf [1] Numerical Methods for Conseration Laws. 5th ed. Berlin:Birkhäuser Verlag. R LeVeque,