Source code for pysph.sph.solid_mech.basic

"""
Basic Equations for Solid Mechanics
###################################

References
----------
.. [Gray2001] J. P. Gray et al., "SPH elastic dynamics", Computer Methods 
    in Applied Mechanics and Engineering, 190 (2001), pp 6641 - 6662.
"""

from pysph.sph.equation import Equation
from textwrap import dedent

[docs]class MonaghanArtificialStress(Equation): r"""**Artificial stress to remove tensile instability** The dispersion relations in [Gray2001] are used to determine the different components of :math:`R`. Angle of rotation for particle :math:`a` .. math:: \tan{2 \theta_a} = \frac{2\sigma_a^{xy}}{\sigma_a^{xx} - \sigma_a^{yy}} In rotated frame, the new components of the stress tensor are .. math:: \bar{\sigma}_a^{xx} = \cos^2{\theta_a} \sigma_a^{xx} + 2\sin{\theta_a} \cos{\theta_a}\sigma_a^{xy} + \sin^2{\theta_a}\sigma_a^{yy}\\ \bar{\sigma}_a^{yy} = \sin^2{\theta_a} \sigma_a^{xx} + 2\sin{\theta_a} \cos{\theta_a}\sigma_a^{xy} + \cos^2{\theta_a}\sigma_a^{yy} Components of :math:`R` in rotated frame: .. math:: \bar{R}_{a}^{xx}=\begin{cases}-\epsilon\frac{\bar{\sigma}_{a}^{xx}} {\rho^{2}} & \bar{\sigma}_{a}^{xx}>0\\0 & \bar{\sigma}_{a}^{xx}\leq0 \end{cases}\\ \bar{R}_{a}^{yy}=\begin{cases}-\epsilon\frac{\bar{\sigma}_{a}^{yy}} {\rho^{2}} & \bar{\sigma}_{a}^{yy}>0\\0 & \bar{\sigma}_{a}^{yy}\leq0 \end{cases} Components of :math:`R` in original frame: .. math:: R_a^{xx} = \cos^2{\theta_a} \bar{R}_a^{xx} + \sin^2{\theta_a} \bar{R}_a^{yy}\\ R_a^{yy} = \sin^2{\theta_a} \bar{R}_a^{xx} + \cos^2{\theta_a} \bar{R}_a^{yy}\\ R_a^{xy} = \sin{\theta_a} \cos{\theta_a}\left(\bar{R}_a^{xx} - \bar{R}_a^{yy}\right) """ def __init__(self, dest, sources, eps=0.3): r""" Parameters ---------- eps : float constant """ self.eps = eps super(MonaghanArtificialStress, self).__init__(dest, sources) def _cython_code_(self): code = dedent(""" cimport cython from pysph.base.linalg3 cimport eigen_decomposition from pysph.base.linalg3 cimport transform_diag_inv """) return code
[docs] def loop(self, d_idx, d_rho, d_p, d_s00, d_s01, d_s02, d_s11, d_s12, d_s22, d_r00, d_r01, d_r02, d_r11, d_r12, d_r22): r"""Compute the stress terms Parameters ---------- d_sxx : DoubleArray Stress Tensor Deviatoric components (Symmetric) d_rxx : DoubleArray Artificial stress components (Symmetric) """ # 1/rho_a^2 rhoi = d_rho[d_idx] rhoi21 = 1./(rhoi * rhoi) ## Matrix and vector declarations ## # Matrix of Eigenvectors (columns) R = declare('matrix((3,3))') # Artificial stress in the original coordinates Rab = declare('matrix((3,3))') # Stress tensor with pressure. S = declare('matrix((3,3))') # Eigenvalues V = declare('matrix((3,))') # Artificial stress in principle direction rd = declare('matrix((3,))') # get the diagonal terms for the stress tensor adding pressure S[0][0] = d_s00[d_idx] - d_p[d_idx] S[1][1] = d_s11[d_idx] - d_p[d_idx] S[2][2] = d_s22[d_idx] - d_p[d_idx] S[1][2] = S[2][1] = d_s12[d_idx] S[0][2] = S[2][0] = d_s02[d_idx] S[0][1] = S[1][0] = d_s01[d_idx] # compute the principle stresses eigen_decomposition(S, R, cython.address(V[0])) # artificial stress corrections if V[0] > 0: rd[0] = -self.eps * V[0] * rhoi21 else : rd[0] = 0 if V[1] > 0: rd[1] = -self.eps * V[1] * rhoi21 else : rd[1] = 0 if V[2] > 0: rd[2] = -self.eps * V[2] * rhoi21 else : rd[2] = 0 # transform artificial stresses in original frame transform_diag_inv(cython.address(rd[0]), R, Rab) # store the values d_r00[d_idx] = Rab[0][0]; d_r11[d_idx] = Rab[1][1]; d_r22[d_idx] = Rab[2][2] d_r12[d_idx] = Rab[1][2]; d_r02[d_idx] = Rab[0][2]; d_r01[d_idx] = Rab[0][1]
[docs]class MomentumEquationWithStress2D(Equation): r"""**Momentum Equation with Artificial Stress** .. math:: \frac{D\vec{v_a}^i}{Dt} = \sum_b m_b\left(\frac{\sigma_a^{ij}}{\rho_a^2} +\frac{\sigma_b^{ij}}{\rho_b^2} + R_{ab}^{ij}f^n \right)\nabla_a W_{ab} where .. math:: f_{ab} = \frac{W(r_{ab})}{W(\Delta p)}\\ R_{ab}^{ij} = R_{a}^{ij} + R_{b}^{ij} """ def __init__(self, dest, sources, wdeltap=-1, n=1): r""" Parameters ---------- wdeltap : float evaluated value of :math:`W(\Delta p)` n : float constant with_correction : bool switch for using tensile instability correction """ self.wdeltap = wdeltap self.n = n self.with_correction = True if wdeltap < 0: self.with_correction = False super(MomentumEquationWithStress2D, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_au, d_av): d_au[d_idx] = 0.0 d_av[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_rho, s_rho, s_m, d_p, s_p, d_s00, d_s01, d_s11, s_s00, s_s01, s_s11, d_r00, d_r01, d_r11, s_r00, s_r01, s_r11, d_au, d_av, WIJ, DWIJ): pa = d_p[d_idx] pb = s_p[s_idx] rhoa = d_rho[d_idx] rhob = s_rho[s_idx] rhoa21 = 1./(rhoa * rhoa) rhob21 = 1./(rhob * rhob) s00a = d_s00[d_idx] s01a = d_s01[d_idx] s10a = d_s01[d_idx] s11a = d_s11[d_idx] s00b = s_s00[s_idx] s01b = s_s01[s_idx] s10b = s_s01[s_idx] s11b = s_s11[s_idx] r00a = d_r00[d_idx] r01a = d_r01[d_idx] r10a = d_r01[d_idx] r11a = d_r11[d_idx] r00b = s_r00[s_idx] r01b = s_r01[s_idx] r10b = s_r01[s_idx] r11b = s_r11[s_idx] # Add pressure to the deviatoric components s00a = s00a - pa s00b = s00b - pb s11a = s11a - pa s11b = s11b - pb # compute the kernel correction term if self.with_correction: fab = WIJ/self.wdeltap fab = pow(fab, self.n) art_stress00 = fab * (r00a + r00b) art_stress01 = fab * (r01a + r01b) art_stress11 = fab * (r11a + r11b) else: art_stress00 = 0.0 art_stress01 = 0.0 art_stress11 = 0.0 # compute accelerations mb = s_m[s_idx] d_au[d_idx] += mb * (s00a*rhoa21 + s00b*rhob21 + art_stress00) * DWIJ[0] + \ mb * (s01a*rhoa21 + s01b*rhob21 + art_stress01) * DWIJ[1] d_av[d_idx] += mb * (s10a*rhoa21 + s10b*rhob21 + art_stress01) * DWIJ[0] + \ mb * (s11a*rhoa21 + s11b*rhob21 + art_stress11) * DWIJ[1]
[docs]class HookesDeviatoricStressRate2D(Equation): r""" **Rate of change of stress (2D)** .. math:: \frac{dS^{ij}}{dt} = 2\mu\left(\epsilon^{ij} - \frac{1}{3}\delta^{ij} \epsilon^{ij}\right) + S^{ik}\Omega^{jk} + \Omega^{ik}S^{kj} where .. math:: \epsilon^{ij} = \frac{1}{2}\left(\frac{\partial v^i}{\partial x^j} + \frac{\partial v^j}{\partial x^i}\right)\\ \Omega^{ij} = \frac{1}{2}\left(\frac{\partial v^i}{\partial x^j} - \frac{\partial v^j}{\partial x^i} \right) """ def __init__(self, dest, sources, shear_mod): r""" Parameters ---------- shear_mod : float shear modulus (:math:`\mu`) """ self.shear_mod = shear_mod super(HookesDeviatoricStressRate2D, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_as00, d_as01, d_as11): d_as00[d_idx] = 0.0 d_as01[d_idx] = 0.0 d_as11[d_idx] = 0.0
[docs] def loop(self, d_idx, d_s00, d_s01, d_s11, d_v00, d_v01, d_v10, d_v11, d_as00, d_as01, d_as11): v00 = d_v00[d_idx] v01 = d_v01[d_idx] v10 = d_v10[d_idx] v11 = d_v11[d_idx] s00 = d_s00[d_idx] s01 = d_s01[d_idx] s10 = d_s01[d_idx] s11 = d_s11[d_idx] # strain rate tensor is symmetric eps00 = v00 eps01 = 0.5 * (v01 + v10) eps10 = eps01 eps11 = v11 # rotation tensor is asymmetric omega01 = 0.5 * (v01 - v10) omega10 = -omega01 tmp = 2.0*self.shear_mod trace = 1.0/3.0 * (eps00 + eps11) # S_00 d_as00[d_idx] = tmp*( eps00 - trace ) + \ ( s01*omega01 ) + ( s10*omega01 ) # S_01 d_as01[d_idx] = tmp*(eps01) + \ ( s00*omega10 ) + ( s11*omega01 ) # S_11 d_as11[d_idx] = tmp*( eps11 - trace ) + \ ( s10*omega10 ) + ( s01*omega10 )
[docs]class EnergyEquationWithStress2D(Equation): def __init__(self, dest, sources, alpha=1.0, beta=1.0, eta=0.01): self.alpha = alpha self.beta = beta self.eta = eta super(EnergyEquationWithStress2D,self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_ae): d_ae[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, s_m, d_rho, s_rho, d_p, s_p, d_cs, s_cs, d_ae, XIJ, VIJ, DWIJ, HIJ, R2IJ, RHOIJ1): rhoa = d_rho[d_idx] ca = d_cs[d_idx] pa = d_p[d_idx] rhob = s_rho[s_idx] cb = s_cs[s_idx] pb = s_p[s_idx] mb = s_m[s_idx] rhoa2 = 1./(rhoa*rhoa) rhob2 = 1./(rhob*rhob) # artificial viscosity vijdotxij = VIJ[0]*XIJ[0] + VIJ[1]*XIJ[1] + VIJ[2]*XIJ[2] piij = 0.0 if vijdotxij < 0: cij = 0.5 * (d_cs[d_idx] + s_cs[s_idx]) muij = (HIJ * vijdotxij)/(R2IJ + self.eta*self.eta*HIJ*HIJ) piij = -self.alpha*cij*muij + self.beta*muij*muij piij = piij*RHOIJ1 vijdotdwij = VIJ[0]*DWIJ[0] + VIJ[1]*DWIJ[1] + VIJ[2]*DWIJ[2] # thermal energy contribution d_ae[d_idx] += 0.5 * mb * (pa*rhoa2 + pb*rhob2 + piij)
[docs] def post_loop(self, d_idx, d_rho, d_s00, d_s01, d_s11, s_s00, s_s01, s_s11, d_v00, d_v01, d_v10, d_v11, d_ae): # particle density rhoa = d_rho[d_idx] # deviatoric stress rate (symmetric) s00a = d_s00[d_idx] s01a = d_s01[d_idx] s10a = d_s01[d_idx] s11a = d_s11[d_idx] # strain rate tensor (symmetric) eps00 = d_v00[d_idx] eps01 = 0.5 * (d_v01[d_idx] + d_v10[d_idx]) eps10 = eps01 eps11 = d_v11[d_idx] # energy acclerations sdoteij = s00a*eps00 + s01a*eps01 + s10a*eps10 + s11a*eps11 d_ae[d_idx] += 1./rhoa * sdoteij