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Newtonian N-Body Simulator

Newtonian N-Body Simulator. Michelle Teh, Michael Witten April 27, 2007. Project Overview. *. v2. N-body problem  Masses  Locations  Velocities  Geometries  Other Properties. v1. v3. No closed form. v4. 1. 2. 3. 4. Iterative Solution. *. n objects

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Newtonian N-Body Simulator

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  1. Newtonian N-Body Simulator Michelle Teh, Michael Witten April 27, 2007

  2. Project Overview... * v2 N-body problem Masses Locations Velocities Geometries Other Properties v1 v3 No closed form. v4 Newtonian N-Body Simulator

  3. 1 2 3 4 Iterative Solution... * • n objects • (n-1) interactions • n(n-1) total interactions • Equal and Opposite Forces • ½ n(n-1) full steps: O(n2) 1 : 2, 3, 42 : 3, 43 : 4 Newtonian N-Body Simulator

  4. Parallel Processing... * All 2-Body Interactions calculated simultaneously involves: x ÷ + - √ Multiplication Division Addition Subtraction Square Root • Fixed-point: 6k-10k Slices • Floating (single): 18k Slices • ½(n2 – n) 2-body calculations (approx.) • (6k slices)(½(n2 – n)) = 33k slices = All slices  n = 3.8 objects => 3 objects Newtonian N-Body Simulator

  5. Parallel Processing... * Stage Pipelining • Only one 2-body interaction per cycle (after latency) • Reuse of hardware • Possibility of gigantic latency between frames • Small by human standards • We can perform other calculations  Still better than a computer Newtonian N-Body Simulator

  6. Simulator Design... * Calculate Gravitational Acceleration Apply Accumulative Acceleration Draw the Graphical Data into a Double Buffer Allow User Interaction with the PS/2 Mouse Newtonian N-Body Simulator

  7. Block Diagram... * FSM Acceleration Applier BRAM Accelerations BRAM Center_x BRAM Center_y BRAM Velocity_x BRAM Velocity_y BRAM Mass BRAM Radius Gravitational Calculator Circle Drawer pixelData pixelAddress PS/2 Mouse Double Buffer Screen Display User Newtonian N-Body Simulator

  8. Parallel Processing... * Stage Pipelining 0 : 1 2 3 4 5 1 : 2 3 4 52 : 3 4 53 : 4 5 4 : 5 bubble • Analysis • 0 Primary for all stages (never Secondary) • 1 Primaryfor all except the last stage • nSecondary for all stages (never Primary) • bubbles per stage = floor[stage number ÷ 2] PrimarySecondary n objects = (n-1) stages where n is even Newtonian N-Body Simulator

  9. r vx (x,y) v m vy Data Representation... * • 144 of 512 x 36 BRAMs (cascaded for larger widths) • Each number has its own BRAM. • Single-precision floats (32 bits) • Range of magnitudes • 2 independent, synchronous read/write ports • Limits parallel access to the data. scalar vector Newtonian N-Body Simulator

  10. Graphical Data... * Uses Bresenham’s Circle Drawing Algorithm (Courtesy of http://glasnost.itcarlow.ie/~powerk/Graphics/Notes) Newtonian N-Body Simulator

  11. (Courtesy of http://agumbo.com/logitech_mouse/page5/) User Interaction... * • Move object • Add object • Remove object • Manipulate coordinate system Newtonian N-Body Simulator

  12. Questions & Answers... Newtonian N-Body Simulator

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