A Particle-system Approach to Real-time Non-linear Analysis

The paper describes a project to apply the particle system (PS) approach to structural engineering problems. The PS approach is widely used in computer graphics and animation to simulate physical phenomena such as deformable objects and collisions. The study finds that although the PS approach is less efficient than conventional finite element analysis, its ability to model unstable structures and support real-time interaction give it particular promise in teaching non-linear behavior, and in preliminary non-linear analysis. The paper also presents examples to verify accuracy of the PS approach, including a large deflection cantilever beam and an inelastic two-bay frame. Introduction Computer graphics and animation have long incorporated the modelling of physical phenomena such as projectile motion and elasticity, and recent animation applications have included sophisticated finite element modelling and fracture mechanics (O'Brien 1999). Computer games now incorporate physical modelling where calculations are done in real time, responding to input from the player. The physical phenomena in computer graphics are typically non-linear, such as contact phenomena and large displacements. Thus, many computer games implement calculation engines to perform non-linear time history analysis in real time, and they do it using an analytic approach fundamentally different from traditional structural analysis. One of the most widely-used approaches to physical modelling in computer graphics is a particle system, where time-step simulation is used to solve the differential equations of motion of a collection of particle masses (Witkin 1997). Selected masses are commonly connected by springs, which are essentially elastic truss elements. The particle system (PS) approach differs from conventional structural analysis in that it does not form a global stiffness matrix, allowing it to easily analyze unstable structures. The PS approach requires significantly more computational resources, but also enables new types of analysis and modes of interactivity. The paper describes a project to apply the PS approach to problems of structural and earthquake engineering, to demonstrate and assess its potential uses. The two-dimensional prototype program for the project is called Arcade. Associate Professor, Dept. of Architecture, University of Virginia, P.O. Box 400122, Charlottesville VA 22904 Interface Models and Interactivity Despite enormous gains in speed, capacity, and usability, structural analysis software retains much of the batch-processing organization it had in the 1950s: prepare input, analyze, and interpret output. The tools for preparing input and interpreting output are far more interactive and graphic, and analysis capacity has increased thousands of times, but the three-phase model remains. Computer speed has now reached a point where it is possible to perform analyses in real time; i.e. structural response can be calculated faster that the response occurs. Real time calculation essentially puts the three phases of input, analysis, and output into a continuously running loop. When this loop includes graphic rendering of the structure, it is possible to create a game-like interface where the structural model responds instantly to input from the keyboard and mouse. Figure 1 shows a simple example of the prototype Arcade program, showing the direct manipulation of a cantilever truss structure, modelled with 22 nodes and 41 elements. A node is selected by clicking on it, and then subsequent clicking and dragging of the mouse creates an element connecting the cursor to the selected node. As the mouse is dragged, the cursor effectively pushes and pulls on the structure via the connecting element; the response is calculated and displayed at a frame rate exceeding 70 frames per second on a computer with a 1.4 GHz processor. Figure 1. Truss structure loaded interactively via the mouse. The black outline shows the unloaded configuration. The grey shows steps of loading Figure 2 shows an example of a tension structure modelled as a chain of linear truss elements. The black outline shows the initial catenary configuration when all masses are equal. The grey outlines show the change in configuration with an increase to the mass of one of the nodes, invoked by a keyboard command. Figure 2. Tension structure where the mass of one node has been increased. The black outline shows the catenary configuration where all nodal masses are equal. The grey shows the change in shape. These simple examples give some sense of the interactive nature of the program. The structural model responds instantly to input from the keyboard and mouse, so that input, processing, and output are running continuously in a real-time loop Theory and Numeric Methods Analysis algorithm Figure 3 shows the basic organization of the analysis loop for a particle system. The analysis is a time step simulation where the position and velocity of each particle (i.e. node) are known at the beginning of each step. Based on this position and velocity, forces are calculated for each node. These forces may arise from several sources. The Arcade implementation includes mechanical forces from the distortion of connected elements, viscous damping forces based on node velocity, contact forces resulting from collisions with surfaces, and forces specified as boundary conditions (i.e. applied loads). With the forces on each node known, the acceleration for each degree of freedom is calculated by Newton's second law, dividing the mass associated with the degree of freedom by the corresponding force. With the acceleration for each node known, the velocity and position at the end of the time step can be calculated using numeric methods for ordinary differential equations. The analysis is then ready to repeat the loop for the next time step. position, velocity forces accel. elements