Space Integration

Overall, the equation of motion for translational velocities with anti-hourglass and contact forces is written as:(1)
M v t = F ext F int + F bod + F hgr + F cont + F trm MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHnbWaaS aaaeaacqGHciITcaWG2baabaGaeyOaIyRaamiDaaaacqGH9aqpcaWH gbWaaSbaaSqaaiaadwgacaWG4bGaamiDaaqabaGccqGHsislcaWHgb WaaSbaaSqaaiGacMgacaGGUbGaaiiDaaqabaGccqGHRaWkcaWHgbWa aSbaaSqaaiaadkgacaWGVbGaamizaaqabaGccqGHRaWkcaWHgbWaaS baaSqaaiaadIgacaWGNbGaamOCaaqabaGccqGHRaWkcaWHgbWaaSba aSqaaiaadogacaWGVbGaamOBaiaadshaaeqaaOGaey4kaSIaaCOram aaBaaaleaacaWG0bGaamOCaiaad2gaaeqaaaaa@598B@
Where, F t r m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHgbWaaS baaSqaaiaadshacaWGYbGaamyBaaqabaaaaa@3A3C@ is the transport momentum vector given as F t r m = e l e m e n t s f t r m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHgbWaaS baaSqaaiaadshacaWGYbGaamyBaaqabaGccqGH9aqpdaaeqbqaaiaa hAgadaahaaWcbeqaaiaadshacaWGYbGaamyBaaaaaeaacaWGLbGaam iBaiaadwgacaWGTbGaamyzaiaad6gacaWG0bGaam4Caaqab0Gaeyye Iuoaaaa@48E7@ .(2)
f i I t r m = ( 1 + η I ) v ρ Φ I ( w j v j ) v i x j d V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHMbWaa0 baaSqaaiaadMgacaWGjbaabaGaamiDaiaadkhacaWGTbaaaOGaeyyp a0ZaaeWaaeaacaaIXaGaey4kaSIaeq4TdG2aaSbaaSqaaiaadMeaae qaaaGccaGLOaGaayzkaaGaeyyXIC9aa8quaeaacqaHbpGCcqGHflY1 cqqHMoGrdaWgaaWcbaGaamysaaqabaGcdaqadaqaaiaadEhadaWgaa WcbaGaamOAaaqabaGccqGHsislcaWG2bWaaSbaaSqaaiaadQgaaeqa aaGccaGLOaGaayzkaaGaeyyXIC9aaSaaaeaacqGHciITcaWG2bWaaS baaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWG QbaabeaaaaGccaWGKbGaamOvaaWcbaGaamODaaqab0Gaey4kIipaaa a@60F8@

All matrices and vectors defined in the above equation are integrated over the spatial domain. The mass matrix is not constant in time, since the density and domain vary with time. It is shown that the solution of the equation is oscillatory in space when a mesh parameter, known as the Peclet number exceeds a critical value. This spatial stability can be avoided by adding numerical diffusion to the scheme which otherwise is generally under-diffusive and thus unstable. The momentum upwind coefficient is defined in material input. The full upwind case (coef=1) is the default value in Radioss and is generally used.

One upwinding technique is available in Radioss:
  • Streamline Upwinding by Petrov-Galerkin method (SUPG):
    SUPG consists in modifying the shape functions to take into account the momentum convection terms. Since version 2018, SUPG is activated by default. SUPG can be turned off by using the Engine file option:
    /UPWM/SUPG