Maps & Tents. Jay Gopalakrishnan. Portland State University. in collaboration with P. Monk, J. Schöberl, P. Sepúlveda, C.

Maps & Tents Jay Gopalakrishnan Portland State University in collaboration with P. Monk, J. Schöberl, P. Sepúlveda, C. Wintersteiger IMA, Minneapolis, June 2017 Jay Gopalakrishnan 1/46

Who are we celebrating? I am the one on the left! [Source: www-users.math.umn.edu/ bcockbur/] Jay Gopalakrishnan 2/46

His 4 golden rules FUN NEW DOABLE IMPORTANT Jay Gopalakrishnan 3/46

Outline Introduction to tent pitching Basic idea History & background A one-dimensional example One-dimensional wave equation New mathematics for tent spaces General multidimensional hyperbolic systems Multidimensional tents Advantages & drawbacks Introducing MTP schemes Maps Examples: Wave, Maxwell & Euler systems Jay Gopalakrishnan 4/46

Standard time stepping 1 Discretize all spatial differential operators. 2 Time step explicitly or implicitly using a time discretization. t Jay Gopalakrishnan 5/46

Standard time stepping 1 Discretize all spatial differential operators. 2 Time step explicitly or implicitly using a time discretization. t Goal of this talk: Explore an alternate time advance mechanism for hyperbolic problems. Jay Gopalakrishnan 5/46

Spacetime tents Suppose we want to solve a hyperbolic problem in one space dimension, with finite propagation speed. t light cone initial data given x Jay Gopalakrishnan 6/46

Spacetime tents Suppose we want to solve a hyperbolic problem in one space dimension, with finite propagation speed. t tent pole initial data given Pitch a tent x Jay Gopalakrishnan 6/46

Spacetime tents Suppose we want to solve a hyperbolic problem in one space dimension, with finite propagation speed. t Pitch a tent initial data given x Problem is solvable in the tent provided the tent pole is not too high. Jay Gopalakrishnan 6/46

Spacetime tents Suppose we want to solve a hyperbolic problem in one space dimension, with finite propagation speed. t initial data given x Problem is solvable in the tent provided the tent pole is not too high. Jay Gopalakrishnan 6/46

Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t x Jay Gopalakrishnan 7/46

Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t t = 1 c (c = maximal characteristic speed) x Jay Gopalakrishnan 7/46

Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ t = 1 c (c = maximal characteristic speed) Local CFL-condition: τ < 1 c x Jay Gopalakrishnan 7/46

Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46

Ancestry of Tent Pitching Ideas to advance in time by local operations in spacetime regions: [Richter 1994] Explicit elements using DG for wave eq. [Lowrie+Roe+van Leer 1995] Spacetime discontinuous Galerkin (SDG) method [Falk+Richter 1999] Explicit DG elements for some Friedrichs systems [Yin+Acharia+Sobh+Haber+Tortorelli 2000] SDG method for elastodynamics [Üngör+Sheffer 2000] Meshing by pitching tents in spacetime [Palaniappan+Haber+Jerrard 2004] SDG for conservation laws [Erickson+Guoy+Sullivan+Üngör 2005] Improved Tent Pitcher algorithm [Monk+Richter 2005] Wave equation in 2D using tent-pitched meshes and SDG Jay Gopalakrishnan 8/46

Wave operator Wave equation: Let K R 2 be a spacetime domain. c 2 tt φ xx φ = f, on K. Rewrite: Set u 1 = c x φ and u 2 = t φ. [ ] [ ] [ ] [ ] t u 1 0 c x u 1 0 = t u 2 c 0 x u 2 c 2. f }{{}}{{} Au F Wave operator: A [ u1 u 2 A is a first order hyperbolic operator. ] [ ] t u = 1 c x u 2. t u 2 c x u 1 Jay Gopalakrishnan 10/46

The wave problem on a tent We want to solve the wave equation using tent pitching. The spacetime tent K can be of various shapes: (interior) K (left boundary) K K (right boundary) On each tent, we must discretize and solve Au = f the wave equation on a tent K, u g V with boundary & initial conditions (b.i.c.). Here V is the correct space with homogeneous b.i.c. Jay Gopalakrishnan 11/46

Space and trace Operator: A [ u1 u 2 ] [ ] t u = 1 c x u 2. t u 2 c x u 1 Graph space: Modern Friedrichs theory applies for A. [Jensen 2004], [Ern+Guermond+Caplain 2007] W = {v L 2 (K) 2 : Av L 2 (K) 2 }. Problem formulation: Au = f on a spacetime tent K, u g V b.i.c. on K. To say what V is, we need trace theory for W. Jay Gopalakrishnan 12/46

Diagonalize & simplify To understand the wave operator [ ] [ ] [ ] u1 0 c u1 Au = t u 2 c 0 x u 2 [ ] [ ] u1 = t QΛQ t u1 u x, 2 u 2 Here Λ = [ ] c 0, 0 c Q = 1 2 [ 1 1 1 1 it suffices to understand the diagonalized (decoupled) operator D = Q t AQ: [ ] [ ] [ ] w1 D = Q t w1 tw 1 + c xw 1 AQ =. w 2 w 2 tw 2 c xw 2 Simplification! It suffices to study the graph space of the advection operator acting on scalar functions w. Âw = t w + α x w ]. Jay Gopalakrishnan 13/46

Advection operator The advection operator Â = t + α x is also a Friedrichs operator. Let its graph space be Ŵ. Traces of Ŵ were studied earlier. [Bardos 1970], [Ern+Guermond 2004] Known trace result for advection If w is in Ŵ and dist(γ i, Γ o ) > 0, then the inflow and outflow traces w Γi, w Γo are in L 2 (Γ i ) and L 2 (Γ o ), respectively. inflow Γ i K outflow Γ o advection vector Jay Gopalakrishnan 14/46

Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. For scalar u on tents K, define Âu = u d t + u α i. x i i=1 Jay Gopalakrishnan 15/46

Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ i Γ i For scalar u on tents K, define Âu = u d t + u α i. x i i=1 Inflow boundary Γ i = part of K where n t + α n x < 0. Jay Gopalakrishnan 15/46

Advection on tents Γ o The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ o For scalar u on tents K, define Âu = u d t + u α i. x i i=1 Inflow boundary Γ i = part of K where n t + α n x < 0. Outflow boundary Γ o = part of K where n t + α n x > 0. Jay Gopalakrishnan 15/46

Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Γ io For scalar u on tents K, define Âu = u d t + u α i. x i i=1 Inflow boundary Γ i = part of K where n t + α n x < 0. Outflow boundary Γ o = part of K where n t + α n x > 0. Define Γ io = Γ i Γ o and δ(z) = dist(z, Γ io ). Jay Gopalakrishnan 15/46

Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x =0.36 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46

Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x =0.25 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46

Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x =0.18 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46

Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x =0.1 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46

Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x =0 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46

Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x = 0.04 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46

Trace theorem for advection on tents Let Âu = u d t + u α i and let Ŵ be the graph space of Â. x i i=1 For smooth functions w on a tent K, let τ i w = w Γi, τ o w = w Γo. Theorem [G+Monk+Sepúlveda 2015] Suppose Assumption [T] holds. Then, the maps τ i and τ o extend to continuous linear operators Moreover, are continuous. τ i : Ŵ L 2 δ (Γ i) and τ o : Ŵ L 2 δ (Γ o). τ i : ker(τ o ) L 2 1/δ (Γ i) and τ o : ker(τ i ) L 2 1/δ (Γ o) Jay Gopalakrishnan 16/46

Trace theorem for advection on tents Let Âu = u d t + u α i and let Ŵ be the graph space of Â. x i i=1 Interpreting τ i : Ŵ L 2 δ (Γ i) continuity: Inflow trace τ i w may behave badly near Γ io because we can only expect Γ i δ τ i w 2 <. Γ io Γ i Γ io δ 0 Jay Gopalakrishnan 16/46

Trace theorem for advection on tents Let Âu = u d t + u α i and let Ŵ be the graph space of Â. x i i=1 Interpreting τ i : ker(τ o ) L 2 1/δ (Γ i) continuity: If w Γo = 0, then to satisfy (τ i w)(x) 0 weakly as x Γ io, Γ i 1 δ τ iw 2 <. Γ io 00000000000000000000000000 Γ i 00000000000000000000 Γ io δ 0 Jay Gopalakrishnan 16/46

Summary of corollaries for the wave operator Traces. For any w W, the graph space of wave operator A, the trace w tent bottom exist in a weighted Lebesgue space. Define V = {w W : w tent bottom = 0}. Wellposedness. Friedrichs theory = A : V L 2 is a bijection. Our tent problem is uniquely solvable for any f L 2. { Au = f on K, u V, b.i.c. on K. Discretization. Continuous Lagrange finite element space satisfies the weak continuity requirements at Γ io. A stable test space generates a Petrov Galerkin discretization on tents. [G+Monk+Sepúlveda 2015] Jay Gopalakrishnan 17/46

Uniform grid error analysis On uniform grids, the method (after condensation) reduces to the leap frog scheme for systems. [ U n+1] j V n+1 j tent [ U n j 1 V n j 1 ] [ U n ] j+1 V n j+1 [ U n 1] j V n 1 j h/2 k/2 Von Neumann analysis shows second order convergence whenever Assumption [T] holds. Jay Gopalakrishnan 18/46

His 4 golden rules FUN NEW! DOABLE IMPORTANT Jay Gopalakrishnan 19/46

Next Introduction to tent pitching....................................! Basic idea.......................................................! History & background...........................................! A one-dimensional example.....................................! One-dimensional wave equation..................................! New mathematics for tent spaces.................................! General multidimensional hyperbolic systems Multidimensional tents Advantages & drawbacks Introducing MTP schemes Maps Examples: Wave, Maxwell & Euler systems Jay Gopalakrishnan 20/46

The problem A general hyperbolic problem Let Ω 0 R N be a spatial domain. On the spacetime domain Ω = Ω 0 (0, T ), we are given smooth functions f : Ω R L R L N, g : Ω R L R L, b : Ω R L R L. Find a function u : Ω R L satisfying t g(x, t, u(x, t)) + div xf (x, t, u(x, t)) + b(x, t, u(x, t)) = 0 where div x is the spatial (x) divergence applied row-wise. Assume that this system of L equations is hyperbolic in the t-direction. Jay Gopalakrishnan 21/46

Tent meshing a time slab Solvable in parallel: Initial level 1 (red) tents Jay Gopalakrishnan 22/46

Tent meshing a time slab Solvable in parallel: Level 2 (orange) tents Jay Gopalakrishnan 22/46

Tent meshing a time slab Solvable in parallel: Level 3 (yellow) tents Jay Gopalakrishnan 22/46

Tent meshing a time slab Solvable in parallel: Level 4 tents Jay Gopalakrishnan 22/46

Tent meshing a time slab Solvable in parallel: Level 5 tents Jay Gopalakrishnan 22/46

Tent meshing a time slab Solvable in parallel: Level 6 tents (completing a time slab) Jay Gopalakrishnan 22/46

Tent meshing a time slab Solvable in parallel: Level 6 tents (completing a time slab) Next, repeatedly stack this mesh in time Jay Gopalakrishnan 22/46

Advantages A rational way to incorporate high order approximations, spatial adaptivity, and locally varying time steps, even on complex structures. Jay Gopalakrishnan 23/46

Advantages A rational way to incorporate high order approximations, spatial adaptivity, and locally varying time steps, even on complex structures. Tent pole height restriction is a local CFL constraint. In contrast, in standard timestepping, time step is constrained by the global CFL measure minimal mesh size (maximal degree) 1 2 wave speed Task parallelism is possible, even when widely varying time steps are used at different spatial locations. In contrast, standard timestepping requires delicate synchronization steps when using local time stepping. Jay Gopalakrishnan 23/46

How to improve? t K t All spacetime unknowns within a tent are coupled. x In traditional time stepping, only spatial unknowns are coupled. x The separation of time and space in traditional time stepping avoided excessive coupling of degrees of freedom. Jay Gopalakrishnan 24/46

A new twist: Maps Spacetime Tent Spacetime Cylinder Φ t K v ˆt ˆKˆKˆKˆKˆK v x v ˆx Instead of solving on tent K v, solve after pulling back to a domain where time and space discretizations can be easily separated. Jay Gopalakrishnan 25/46

Introducing MTP Schemes Mapped Tent Pitching (MTP) schemes have these steps: 1 Construct spacetime mesh using a tent pitching (meshing) algorithm. 2 Map conservation law on each tent to a spacetime cylinder. 3 Spatially discretize on the cylinder using your favorite method. 4 Apply explicit or implicit high order time stepping within the cylinder. 5 Map computed solution on the cylinder back to the tent. Jay Gopalakrishnan 26/46

Form of the (Duffy-like) map The map is ) ( ) ˆx Φ = (ˆxˆt ϕ(ˆx, ˆt) where ϕ is defined as follows: K v Φ ˆK v = Ω v (0, 1) t ˆt v Ω v Jay Gopalakrishnan 27/46

Form of the (Duffy-like) map The map is ) ( ) ˆx Φ = (ˆxˆt ϕ(ˆx, ˆt) where ϕ is defined as follows: If tent bottom is the graph of ϕ b, t v K v Ω v Φ ˆt ˆK v = Ω v (0, 1) Jay Gopalakrishnan 27/46

Form of the (Duffy-like) map The map is ) ( ) ˆx Φ = (ˆxˆt ϕ(ˆx, ˆt) where ϕ is defined as follows: If then tent bottom is the graph of ϕ b, tent top is the graph of ϕ t, ϕ = (1 ˆt)ϕ b + ˆtϕ t. t v K v Ω v Important weight: δ = ϕ t ϕ b. Φ ˆt ˆK v = Ω v (0, 1) Jay Gopalakrishnan 27/46

Pull back of the conservation law Let ˆf (w) = f (Φ(ˆx, ˆt), w), ĝ(w) = g(φ(ˆx, ˆt), w), ˆb(w) = b(φ(ˆx, ˆt), w), Ĝ(w) = ĝ(w) ˆf (w) gradˆx ϕ. (To simplify notation, suppress dependence of Ĝ, ˆf, ˆb on ˆx, ˆt.) Theorem [G+Schöberl+Wintersteiger 2016] A function u : K v R L satisfies the hyperbolic equation t g(u) + div xf (u) + b(u) = 0 in K v, if and only if û = u Φ satisfies, with δ = ϕ t ϕ b, ( Ĝ(û) + divˆx δ ˆf (û) ) + δ ˆb(û) = 0 in ˆK v. ˆt Idea of proof: Write the equation using the spacetime divergence operator div xt F + b = 0, apply the Piola map, and manipulate. Jay Gopalakrishnan 28/46

Next Introduction to tent pitching....................................! Basic idea.......................................................! History & background...........................................! A one-dimensional example One-dimensional wave equation..................................! New mathematics for tent spaces.................................! General multidimensional hyperbolic systems Multidimensional tents...........................................! Advantages & drawbacks........................................! Introducing MTP schemes......................................! Maps...........................................................! Examples: Wave, Maxwell & Euler systems Jay Gopalakrishnan 29/46

Wave equation Classical form of the wave problem Find φ : Ω R s.t. tt φ + β t φ div x (αgrad x φ) = 0 in Ω := Ω 0 (0, T ), n x αgrad x φ = 0 on Ω 0 (0, T ) φ = φ 0 on Ω 0 t φ = φ 1 on Ω 0 Jay Gopalakrishnan 30/46

Wave equation Classical form of the wave problem Find φ : Ω R s.t. tt φ + β t φ div x (αgrad x φ) = 0 in Ω := Ω 0 (0, T ), n x αgrad x φ = 0 on Ω 0 (0, T ) φ = φ 0 on Ω 0 t φ = φ 1 on Ω 0 [ ] [ ] q αgradx φ Reformulate as a first order system using = µ t φ [ ] [ ] [ ] [ ] [ ] [ ] α 1 0 q 0 gradx q 0 0 q t + = 0 0 1 µ div x 0 µ 0 β µ }{{}}{{}}{{} g(x,t,u) div x f (u) b(x,t,u) : Jay Gopalakrishnan 30/46

An MTP scheme for the wave equation Equation to solve after mapping to cylinder (apply the theorem): [ ˆα 1 ] ] [ ] [ ] gradˆx ϕ [ˆq ˆt (gradˆx ϕ) t gradˆx (δˆµ) 0 + 1 ˆµ divˆx (δˆq) δ ˆβ ˆµ }{{}}{{}}{{} Ĝ(û) divˆx δˆf (û) δˆb(û) = 0 Use the order p Raviart-Thomas pair of spaces (for ˆq and ˆµ) for the spatial (ˆx) discretization within the cylinder. Use a p-stage implicit Runge-Kutta method for time discretization within the cylinder. Jay Gopalakrishnan 31/46

Wave solution by locally implicit MTP scheme Use MTP scheme (β = 0) to approximate the standing wave φ = cos(πx 1 ) cos(πx 2 ) sin( πt 2 ) 2π. Implementation in NGSolve. Observed O(h p ) convergence. Jay Gopalakrishnan 32/46

His 4 golden rules FUN NEW DOABLE! IMPORTANT Jay Gopalakrishnan 33/46

Doable in 4D! Unified implementation for spatial DIM=2 and DIM=3 possible. 4D tents only need 3D spatial mesh data and tentpole heights. Jay Gopalakrishnan 34/46

3D Maxwell Resonator Spatial mesh is refined along sharp inner curves. MTP scheme automatically adapts local time steps. [Hochbruck+Paur+Schulz+Thawinan+Wieners 2015] Jay Gopalakrishnan 35/46

3D Maxwell Resonator Gaussian electric field pulse used to start simulation. 30 million dofs Hy component of the solution Jay Gopalakrishnan Simulated up to t=260 s with p = 2 in 20 min on E7-8867 4 16 cores 35/46

His 4 golden rules FUN NEW DOABLE IMPORTANT! Jay Gopalakrishnan 36/46

For the first time, we can: 1 Perform explicit time stepping through unstructured tent meshes. 2 Use standard spatial discretizations within tents. Jay Gopalakrishnan 37/46

Data locality example Example: Scalar hyperbolic problem with p = 3 in 3 space dim. We have 20 spatial DG shape functions per spatial tetrahedron. Assume 20 simplices form a 4D spacetime tent. Time propagation operator acts on 400 unknowns. Matrix-free implementation of MTP schemes need only load into memory 400 double precision numbers, or 400 8 3KB. 3KB easily fits into local cache. = Timesteps within a tent require little memory movement. Jay Gopalakrishnan 38/46

Euler equations Euler equations modeling an ideal gas: N = 2, L = 5, with t g(u) + div xf (u) + b(u) = 0, ρ m u = m, g(u) = u, f (u) = PI + m m/ρ, b = 0, E (E + P)m/ρ where the pressure P is related to the state variables by P = 1 2 ρt, T = 4 ( E 5 ρ 1 m 2 ) 2 ρ 2. ρ : Ω R m : Ω R N E : Ω R density momentum total energy Jay Gopalakrishnan 39/46

Wind tunnel Well-known test problem: Mach 3 wind tunnel with a flat-faced step. [Emery 1968] [Woodward+Colella 1984] x 2 1 reflect inflow outflow 0.2 reflect reflect reflect 0 0 0.6 3 x 1 At t = 0: ρ = 1.4, m = ρ [ 3 0 ] t, P = 1. Jay Gopalakrishnan 40/46

Historical computations i ++*. * c. * %. -.. * I * I I,--/ f *- l FIG. 11. Isobars (P/PJ for two-dimensional step as given by Lax-Wendroff method: o = 0.7, n = 1397, ndt = 238.8 Ax/c1 (+ = 10, d = 11) (K was changed from 2.0 to 4.0 at nat = 150.9 Ax/c,). + 8. +. ;t. 21 i -.. * * c-.a*, * z<n -. -0. -. I. -. s-:n -c 8 -.* -* a-89. --..-* *--a [Emery 1968] l 5 l l. + r... a z *.. -.-fl.. I....I (L * * +,r..: -n....#., l.. l c.*.i.. - * -.* c..#. * -4 - -.? l -.. c-s NC *. =; ;,;. - -. -. -m...:*-* --, l -.- *. *. *... l -..... -... l -0. *. -. m. 0 -..- -.... l.. *. -. l * - 8-9 * l I. c.-. * -w* I - --r * l * I ; l * -* * I. 1.*, * Jay Gopalakrishnan 41/46

Historical computations [Woodward+Colella 1984] FIG. 3-Continued. Jay Gopalakrishnan 41/46

Tent pitched time slab Jay Gopalakrishnan 42/46

Results MTP makes it possible to use standard spatial DG discretization on tents. t=4, degree 4 spatial DG, 3951 triangles [Video of time steps] Jay Gopalakrishnan 43/46

Ongoing work: Error analysis Analysis following the technique of [Monk+Richter 2005]: Split spacetime mesh into layers of tents. Analyze error propagation within each layer. Reduce the problem to analysis of the error in a single tent. Jay Gopalakrishnan 44/46

Ongoing work: Improved timestepping Tent system to solve after mapping: Reduction to ODE: Put V = MU. Then ( Ĝ(û) + divˆx δ ˆf (û) ) + δ ˆb(û) = 0 in ˆK v. ˆt d dˆt [MU](ˆt) + AU(ˆt) = 0 d dˆt V + AM 1 V = 0 The presence of M 1 prevents straightforward error analysis. Taylor time stepping preserves polynomials, so holds promise. Jay Gopalakrishnan 45/46

Conclusion New mathematics needed for understanding tent spaces. [G+Monk+Sepúlveda 2015] A tent pitching scheme motivated by Friedrichs theory MTP schemes provide a spacetime discretization that is high order in space and time, is constrained by a local (not global) CFL condition, has new efficiencies due to mapping and decoupling, gives explicit schemes on unstructured spacetime meshes, allows the use of standard DG methods on tents. [G+Schöberl+Wintersteiger 2017] Mapped tent pitching schemes for hyperbolic systems Happy birthday Bernardo! Jay Gopalakrishnan 46/46