Macroscopic models for pedestrian flow and meshfree particle methods

Transcription

1 Macroscopic models for pedestrian flow and meshfree particle methods Axel Klar TECHNISCHE UNIVERSITÄT KAISERSLAUTERN Fachbereich Mathematik Arbeitsgruppe Technomathematik

2 In cooperation with R. Borsche, S. Göttlich, N. Mahato, L. Müller, F. Schneider, S. Tiwari Content Hierarchy of models Numerical method Numerical results Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 2 / 40

3 Hierarchy of models Model hierarchies Fine to coarse scale models microscopic models: interacting particles, social force models (Helbing et al.), vision-based models (Degond et. al. ),... kinetic models: Vlasov-type, Vlasov-McKean equations macroscopic models: macroscopic models with non-local terms (Colombo et. al.) localized models (Hughes et al.,...) Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 3 / 40

4 Hierarchy of models Microscopic models: empirical distributions ρ N δ S (x) = δ S (x x j ) N j Function δ S = δ S (x), x R d with δ S (x) = S d δ ( x S ), δ S (x)dx =. Approximation of Dirac δ = δ(x) as S goes to 0. δ S smooth, rotationally symmetric, monotone decaying Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 4 / 40

5 Hierarchy of models Interaction potentials Function U R = U R (x), x R d. Example: U R smooth, rotationally symmetric, monotone decaying ( repulsive interaction potential). We use again U R (x) = U R d ( x R ) such that U (x)dx = and U R approximates δ for R 0. Joint interaction potential: ρ N U R (x) = U R (x x j ) N j Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 5 / 40

6 Hierarchy of models Interaction potentials for traffic / pedestrian flow unsymmetric, one-sided potentials, vision cones. d = : potentials depending on downstream traffic density with U(x) = 0, x > 0. This gives U(x i x j ) = 0 if x i x j > 0, i.e. x i > x j, i.e. interaction only with predecessor. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 6 / 40

7 Hierarchy of models Microscopic equations for interacting particle system (x i, v i )(t) R 2 R 2, i =,..., N, V (ρ) = ρ. with dx i = v i dt dv i = V (ρ N δ S (x i ))ê(x i )dtdt x ρ N U R (x i )dt γv i dt + AdW i t ê(x) = φ(x) φ(x) where φ is the solution of the eikonal equation φ(x) = ( V (ρ N δ S (x))). Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 7 / 40

8 Hierarchy of models Remark: Reduced microscopic equations Simplified equations neglecting time dependence of velocities for x i (t) R d, i =,..., N, dx i = V (ρ N δ S (x i ))ê(x i )dt x ρ N U R (x i )dt + AdW i t Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 8 / 40

9 Hierarchy of models Mean field limit: Empirical measure The empirical measures of the stochastic processes (x i, v i ) are given by δ N (xi,v i )(x, v). i The mean field limit describes the convergence as N towards a deterministic distribution f = f (x, v). This gives the convergence of ρ N δ R (x i ) = δ R (x i x j ) N to a coarse grained density δ R (x y)ρ(y)dy = δ R ρ(x) j with ρ(y) = f (y, w)dw. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 9 / 40

10 Hierarchy of models Mean field limit Starting from the microscopic equations for (x i, v i )(t), i =,..., N, this gives formally a limit stochastic process (x, v) R 2 R 2, the McKean -Vlasov equation dx = vdt dv = V (δ S ρ)êdt x U R ρdt γvdt + AdW t where ρ(y) = f (y, w)dw and f (x, v) is the distribution of the stochastic process (x, v). Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 0 / 40

11 Hierarchy of models Kinetic mean field equation The corresponding differential equation for the evolution of the distribution functions f = f (x, v, t), x, v R d is given by with force term and diffusion term t f + v x f = Sf + Lf Sf = v (V (δ S ρ)êf ) + v ( x U R ρf ). ) Lf = v (γvf + A2 2 v f. Rem.: For the reduced microscopic problems we obtain for ρ = ρ(x, t) ) t ρ = x (ρ ( V (δ S ρ)ê + x U R ρ) + A2 2 xρ. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods / 40

12 Hierarchy of models References / Lecture notes F. Golse, On the Dynamics of Large Particle Systems in the mean-field limit Golse, F.: The mean field limit for the dynamics of large particle systems. Journées équations aux dérivées partielles, 9 (2003), pp. 47. M. Hauray and P.E. Jabin, Particle approximation of Vlasov equations with singular forces: Propagation of chaos. P.E. Jabin, A review of the mean field limit for the Vlasov equation Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 2 / 40

13 Hierarchy of models Balance equations Multiplying the mean field equation with and v one obtains t ρ + x (ρu) = 0, t ρu + x v vfdv = γρu + ρv (δ S ρ)ê ρ x U R ρ with the momentum ρu(x, t) := vf (x, v, t)dv. Closure problem: approximate v vfdv using ρ and u. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 3 / 40

14 Hierarchy of models The closure problem Solution: choose ansatz function F = F [ρ, u](v) with Fdv = ρ and vfdv = ρu such that f F. Then, v vfdv v vfdv A monokinetic closure: F (v) = ρδ(v u) such that v vfdv = ρu u. Maxwellian closure with variance θ: F = M[ρ, u, θ](v) such that v vfdv = θρi + ρu u. A linear closure: choose F = ρ( + u v) M with for example M(v) = M[, 0, ](v)) such that v vfdv = ρi. F might be negative! Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 4 / 40

15 Hierarchy of models Other closures Higher order expansions matching more moments: ( n ) F (v) = w i v i M i=0 Nonlinear (Maximum-entropy) closures: positive ansatz function F (v) = exp( w i v i ), i=0 for example F (v) = a exp(b v) C.D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys. 83 (996) A. M. Anile, S. Pennisi, and M. Sammartino, A thermodynamical approach to Eddington factors, J. Math. Phys. 32, 544 (99). Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 5 / 40

16 Hierarchy of models Hydrodynamic macroscopic models One obtains (maxwellian closure with variance θ) t ρ + x (ρu) = 0, t ρu + x (ρu u) + θ x ρ = γuρ + ρv (δ S ρ)ê ρ x U R ρ with the momentum ρu(x, t) := vf (x, v, t)dv. ê(x) = φ(x) φ(x) where φ is the solution of the eikonal equation φ(x) = V (δ S ρ(x)). Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 6 / 40

17 Hierarchy of models Localized models δ S or U R approximate for small values of S or R a δ-distribution and one obtains formally hydrodynamic equations of the form t ρ + x (ρu) = 0, t ρu + x (ρu u) = γuρ ρ x ρ + ρv (ρ)ê θ x ρ. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 7 / 40

18 Hierarchy of models Remark: Scalar equations For the reduced microscopic problems we obtain Colombo-type equations t ρ = x ( V (δ s ρ)êρ) + x (ρ U R ρ). The localized versions are Hughes-type models t ρ = x ( V (ρ)êρ) + x (ρ x ρ). together with φ = V (ρ). Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 8 / 40

19 Hierarchy of models Rigorous Work: Nonlocal equations to local equations Numerical investigations: P. Amorim, R. M. Colombo, and A. Teixeira. On the numerical integration of scalar nonlocal conservation laws. ESAIM Math. Model. Numer. Anal., 49():9 37, 205. P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, () (206), Rigorous proof for situations with diffusion A > 0, several counterexample otherwise. Maria Colombo, Gianluca Crippa, Laura V. Spinolo, On the singular local limit for conservation laws with nonlocal fluxes, arxiv S. Blandin and P. Goatin. Well-posedness of a conservation law with non-local flux arising in traffic flow modeling. Numer. Math., 32(2):27 24, 206. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 9 / 40

20 Hierarchy of models Comparison of closures for hydrodynamic pedestrian flow Compare microscopic/kinetic and nonlinear maximum entropy hydrodynamic 0.5 y x Cross-walks: densities at t = 2.4, A = 5 Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 20 / 40

21 Hierarchy of models Comparison of closures for hydrodynamic pedestrian flow Compare microscopic and linear hydrodynamic and scalar y 0 y x Cross-walks: densities at t = 2.4, A = x x Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 2 / 40

22 Hierarchy of models Evacuation time For small A the evacuation time determined from the scalar model differs strongly from the microscopic, the mesoscopic and the hydrodynamic one. For large values of A, all simulations give similar results. 6 5 Micro Mean M Scalar evacuation time A The evacuation time in dependence of the parameter A for microscopic, mesoscopic, nonlinear hydrodynamic and scalar models. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 22 / 40

23 Numerical methods Numerical methods for non-local hydrodynamic equations Arbitrary-Lagrangian-Eulerian particle or moving mesh methods Procedure: Start from lagrangian formulation of hydrodynamic equations dx dt = u dρ dt = ρ x u du dt = γu V (δ S ρ)ê x U R ρ θ ρ xρ. Remark: Lagrangian approach closer to particle idea. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 23 / 40

24 Numerical methods ALE meshfree particle methods Use a smoothing radius ( SPH) and determine derivatives by least square fit on a particle cloud of grid particles x i, i =,, Ñ N! Add a discretization of the convolution integral x U R ρ(x) = x U R (x y)ρ(y)dy Solve resulting system of ODEs particle management (generate /delete grid particles)! Add additional procedures, for example Upwinding / central procedures for hyperbolic problems solution of the eikonal equation on the particle cloud fast marching method for unstructured grids Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 24 / 40

25 Numerical methods Macroscopic particle methods working in the localized limit Problem: relatively small R (in the localized limit) and not very large number of macroscopic particles underresolution Naive/microscopic evaluation of the convolution integral leads to wrong results x U R ρ(x i ) Ñ j=,j i ρ j V j x U R (x i x j ) 0 V j : Voronoi cell around grid particle j with volume V j Very near to a microscopic simulation! Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 25 / 40

26 Numerical methods Macroscopic particle method and localized limit Procedure: Use higher order approximation of the density in the particle method, for example ρ(y) = Ñ [ρ j + σ j (y x j )] χ Vj (y) j= σ j : first order derivative approximated via least square fit from the point cloud plug into convolution integral Compute resulting integrals explicitely ( multiscale finite elements) one obtains correction factors x U R ρ(x i ) ( ) ρ j V j x U R (x i x j ) + σ i α i j i Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 26 / 40

27 Numerical methods Results This leads to a uniform scheme for different R and to the correct localized limit for R 0 even for an underresolved situation. Computation times depends on the number of particles (for microscopic simulation) and grid points (for macroscopic), The particle method can be viewed as a numerical transition from a microscopic model if a very fine resolution is used to a macroscopic model if a coarse resolution and the above fix is used. A. Klar, S. Tiwari, A multi-scale particle method for mean field equations: the general case, SIAM Multiscale Mod. Sim. A. Klar, S. Tiwari, A multi-scale meshfree particle method for macroscopic mean field interacting particle models, SIAM Multiscale Mod. Sim. 204 Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 27 / 40

28 Numerical results Numerical Results Test-cases: I: Conservation laws, shock solutions, Lighthill-Whitham t ρ + x (( U R ρ)ρ) = 0 II: 2D pedestrian dynamics t ρ + x (ρu) = 0 t u + u x u + θ ρ xρ = γ (( U R ρ)ê(x) + x U R ρ u) Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 28 / 40

29 Numerical results Test case I: LWR model Lighthill Whitham solution Naive approximation Multiscale Lighthill Whitham solution Naive approximation Multiscale Lighthill Whitham solution Naive approximation Multiscale 0.9 Lighthill Whitham solution Naive approximation Multiscale Shock solution for N = 800 particles and R = to R = 0.8 for local and non-local model with microscopic and multi-scale approximation and downwind potential. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 29 / 40

30 Numerical results Test case I: LWR model Lighthill Whitham solution Naive approximation Multiscale Lighthill Whitham solution Naive approximation Multiscale Lighthill Whitham solution Naive approximation Multiscale Lighthill Whitham solution Naive approximation Multiscale Shock solution for N = 800 particles and R = to R = 0.8 for local and non-local version with symmetric potential and δ > 0 Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 30 / 40

31 Numerical results Convergence error # particles naive multi-scale CPU time error error in seconds Convergence study for nonlocal Lighthill-Whitham equations with downwind interaction potential and R = 0.2. L 2 error Naive approximation Multiscale No. of particle L 2 -error plot Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 3 / 40

32 Numerical results Test case II: Pedestrian dynamics repulsive interaction potential, Lighthill Whitham type velocity function, coupling to eikonal equation N.K. Mahato, A. Klar, S. Tiwari, Particle methods for multi-group pedestrian flow, Appl. Math. Modeling 53, , 208 R. Etikyala, S. Göttlich, A. Klar, S. Tiwari, Particle methods for pedestrian flow models: from microscopic to non-local continuum models, Mathematical Methods and Models in Applied Sciences 24 (2), , 204 Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 32 / 40

33 Numerical results Pedestrian dynamics: fine resolution N Ñ repulsive potential, coupling to eikonal equation, meshfree solution of the eikonal equation macroscopic microscopic Density plot determined from local limit equation and nonlocal equations (microscopic or multi-scale approximation) for inital spacing x = 0.2 and R = 0.4 (fine resolved situation near local limit) Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 33 / 40

34 Numerical results Pedestrian dynamics: coarse resolution N >> N Density for nonlocal equations with microscopic and multi-scale approximation for x = 0.5, R = 0.2 and local limit.(coarsely resolved situation near local limit) Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 34 / 40

35 Numerical results Evacuation times Normalized density Reference Naive Multiscale Time Time development of the normalized total mass in the computational domain determined from the different models R = 0.2 and coarse initial spacing x = with N = 400 grid particles. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 35 / 40

36 Numerical results Pedestrian dynamics: CPU time initial # particles naive multi-scale CPU time spacing error error min min min min Relative error Naive Multiscale No. of particle 0 4 Comparison of CPU times between microscopic and multiscale simulations. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 36 / 40

37 Numerical results Extensions: Pedestrian dynamics with groups Extension using multiphase / mixture models dx (k) i dt dv (k) i dt = v (k) i M = l= xi U (k,l) ( x (k) i j group (l) x (l) j ) + G (k) (x (k) i, v (k) i, ρ N (x (k) i )), where U (k,l) is an interaction potential denoting the interaction between members of groups k and l. We choose the Morse potential U (k,l) (r) = C a e r/la + C r e r/lr. C a, C r are attractive and repulsive strengths and l a, l r are length scales. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 37 / 40

38 Numerical results Pedestrian dynamics with groups Repulsive potential and Morse potential for attraction in groups Density of pedestrians for single and multi-group hydrodynamic model 0 Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 38 / 40

39 Numerical results Multi-group evacuation times N(t)/N(0) 0.5 Ca = 0, Cr = 200 Ca = 0, Cr = 200 Ca = 50, Cr = 200 Ca = 70, Cr = 200 N(t)/N(0) 0.5 Ca = 0, Cr = 00 Ca = 0, Cr = 00 Ca = 50, Cr = 00 Ca = 70, Cr = Time [s] Time [s] Ratio of initial and actual grid particles over time in single C a = 0 and multi-group C a = 0, 50, 70 hydrodynamic model The evacuation time is larger in the case of grouped pedestrians. Compare experimental results in J. Xi, X. Zou, Z. Chen, J. Huang, Multi-pattern of Complex Social Pedestrian Groups Transportation Research Procedia Volume 2, 204 C. Kruchten, A. Schadschneider, Empirical study on social groups in pedestrian evacuation dynamics, Physica A, 207 Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 39 / 40

40 Numerical results Conclusions Derivation of a hierachy of models for pedestrian flow. The particle method can be viewed as a uniform numerical transition from a microscopic model if a very fine resolution is used to a macroscopic model if a coarse resolution is used. Other interaction models can be included: attraction with center of mass of the group or vision based models M. Moussaid et al., Walking Behaviour of Pedestrian Social Groups and Its Impact on Crowd Dynamics, PLoS ONE P. Degond et al., Vision-based macroscopic pedestrian models, KRM A review of the mathematical aspects: R. Borsche, A. Klar, F. Schneider, Numerical methods for mean field and moment models for pedestrian flow, to appear in Crowd Dynamics Volume - Theory, Models, and Safety Problems, N. Bellomo and L. Gibelli Eds., Birkhäuser-Springer, MSSET Series. Axel Klar (TU Kaiserslautern) Macroscopic models and particle methods 40 / 40