Computational Mechanics Lab
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 1. The development of a new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes. It is based on the minimization of the order of the local truncation error for the discrete or semi-discrete equations and provides the maximum possible order of accuracy compared to other numerical techniques with a similar structure of discrete or semi-discrete equations. Currently, the new technique is applied to the solution of the time dependent and independent elasticity equations, the wave and heat equations, the Poisson equation, the Helmholtz equation. The new approach significantly reduces the computation time at a given accuracy compared to other numerical techniques (e.g., it is 1000 and more times faster than the finite elements and does not require complicated mesh generators for complex irregular domains; see Publications and the Table below).      Table. Improvement of order of accuracy by new approach on 2-D uniform meshes                                                       2-D 9-point stencils for the linear finite elements and 25-point stencils for the quadratic isogeometric elements (similar increase in accuracy is valid in the 3-D case as well) 2. The development of a new two-stage time-integration technique for time integration of structural dynamics and wave propagation problems. This technique includes the stage of basic computations and the separate filtering stage and yields accurate solutions of elastodynamics and acoustic problems. In contrast to existing approaches, the new technique quantifies and filters spurious oscillations without any guesswork. 3. The development of a new exact closed-form a-priori global error estimator in time for the integration of linear elastodynamics problems. This estimator yields the size of time increments Δt for the given tolerance ε of frequency ω (e.g., for the maximum or leading frequency or for the frequency of interest) at the final observation time T :                                   Δt ≤ [(12ε)/(Tω3)]0.5     for implicit and explicit second-order time-integration                                         methods.     E.g, at ε=1% for ω=3π at T=50   →  Δt ≤ 0.00169316.      Δt ≤ [(720ε)/(Tω5)]0.25    for implicit and explicit fourth-order time-integration                                           methods.      E.g, at ε=1% for ω=3π and T=50   →  Δt ≤ 0.0373037 (see IJNME-2011).       Based on the new error estimator, a new simple technique is developed for the increase in accuracy of numerical results at time integration by the trapezoidal rule (this technique with the second-order trapezoidal rule is equivalent to the application high-order time integration methods); see CP-2012 in Publications 4. The development of new finite elements with reduced dispersion error that can be used with explicit and implicit time-integration methods for elastodynamics problems and significantly reduce the computation time compared with that for the standard finite elements. 5. Application of the new accurate numerical approach to the simulation of the propagation of elastic waves in the Split Hopkinson Pressure Bar (SPHB).       Below are shown examples of the application of the new approach to wave propagation in a 3-D irregular domain (using a Cartesian mesh) as well as in a 2-D long elastic cylinder at impact loading (using finite elements). The impact problem cannot be accurately solved by existing numerical methods due to large spurious oscillations.
 Governing Equations Stencils Order of accuracy Order increase with new approach Conventional finite and isogeometric elements New approach 1. Time dependent wave and heat equations 9-points stencils 2 4 4-2=2 25-points stencils 4 8 8-4=4 2. Poisson Equation 9-points stencils 2 4 (rectangular meshes) 6 (square meshes) 4-2=2 (rectangular meshes) 6-2=4 (square meshes) 25-points stencils 4 (rectangular meshes) 6 (square meshes) 14 (rect. meshes) 18 (square meshes) 14-4=10 (rect. meshes) 18-6=12 (square meshes) 3. Time independent Helmholtz equation 9-points stencils 2 4 (rectangular meshes) 6 (square meshes) 4-2=2 (rectangular meshes) 6-2=4 (square meshes) 4. Time dependent elasticity equations 9-points stencils 2 2 0 25-points stencils 4 6 6-4=2 5. Time independent elasticity equations 9-points stencils 2 2 0 25-points stencils 4 10 10-4=6