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The present discussion describes the methodcurrently implemented in EMU. Consider the ratio of the deformed density to the reference density. Thisratio is 1/X. We approximate the ratio of the deformed to reference density at a given node by1X="1VXj r jpj m1Vj#3/m, V =Xk1Vk , (25)where the sum is taken over the nodes inside the horizon of the given node, pj = |η j+ξ j|, r j = |ξ j|,and 1Vj is the reference volume of node j . We refer to the nodes inside the horizon of a given node asits family. Hence, V is the total reference volume of the family of the given node.If m = 1 and the reference-node volumes are equal, (13) implies that the summation in (25) representsthe cube of the average of the inverses of the one plus bond stretches within the family. For m > 0 withconstant bond stretch s, (25) implies that the relative density 1/X is1X= 1(1+s)3. (26)As expected, (26) shows that the density increases in compression (s < 0) and decreases in expansion(s > 0). The purpose of including m = 1 in (25) is to allow for the possibility that bonds of differentlength could sustain different forces even if the deformation is an isotropic expansion. This form canbe helpful, for example, in preventing nodes in a numerical grid from getting so close to each other thatthey overlap.For any m and equal reference-node volumes, (25) becomes1X="1NXj 1(1+s j ) m#3/m, s j = pj −r jr j. (27)Since (27) represents the ratio of deformed to reference density, the more highly compressed bonds(s j < 0) will have a larger contribution to the overall density ratio at a node for m > 1. From Equation (25), the expansion X isX ="1VXj pjr j −m1Vj#−3/m, V =Xk1Vk . (28)From (24) with this approximation for the expansion, the PFF for a gas isfk = 6rkVdWd X pkrk −m−1X1+m/3. (29)If we identify W with the internal energy per unit volume of the gas, then (29) and knowledge of thedependence of W on the expansion X yield an expression for the PFF at node k.From an axiomatic formulation of thermodynamics, such as found in [Callen 1960], the pressure is anintensive variable defined as minus the partial derivative of the internal energy with respect to a specificvolume at constant entropy [Callen 1960, p. 31]. Therefore, the derivative in (29) isdWd X= −P, (30)which implies that the PFF for node k in a gas isfk = − 6PrkV pkrk −m−1X1+m/3. (31)Implementation of (31) requires knowledge of P as a function of X and a value for m. In the currentversion of EMU, a value of m = 1 is used. We have not investigated the consequences of using differentvalues for m or using some alternate formulation of the expansion.For many applications of interest, gases are rapidly expanding. Therefore, in the initial implementationof gas modeling in EMU, gases are treated as ideal gases undergoing an isentropic or adiabatic expansion.In this case, we can relate the pressure required in (31) to the expansion.Consider an isentropic process from a state with volume and temperature (V0, T0) to a state withvolume and temperature (V, T ). The change in entropy, 1S, for an ideal gas between these states isgiven by1S = nR lnVV0+nCV lnTT0, (32)where n is the number of moles of the gas, R is the gas constant (8.31 J/mol/K), and CV is the molarspecific heat at constant volume [Halliday et al. 2001, p. 487]. Setting 1S = 0 in (32) and using theideal gas law (PV = nRT ), we obtain the relationPVγ= P0Vγ0 , (33)where γ = CP/CV is the ratio of molar specific heats and CP = CV + R is the molar specific heat atconstant pressure. Therefore, for an ideal gas, the pressure as a function of the expansion for an adiabaticexpansion is given byP = P0 V0V γ= P0X−γ. (34) 7. The peridynamic detonation model in EMUA peridynamic detonation model was developed and implemented in the EMU computer code. Thefirst component of the detonation model is the input. For each explosive material, the user provides thelocation of the detonation point (xdet) and time of detonation initiation (tdet) along with the density ofthe unreacted explosive (ρun) and the detonation speed (Vdet). Multiple detonation points are permittedin the current version of EMU. For an ideal gas, the user may also specify the pressure (PC J ), ratio ofmolar specific heats (γ ), and detonation temperature (TC J ) of the detonation-product gases. If the userdoes not specify these quantities, they are obtained from the following correlations:PC J = ρun(Vdet)2 1.0−0.7125(ρun/1000)0.04 , (35)γ = 0.7125(ρun/1000) 1.0−0.7125(ρun/1000)0.04 , (36)TC J = 1643γ. (37)The units in (35), (36), and (37) are SI units; the ratio of molar specific heats (γ ) is dimensionless[Cooper 1996, pp. 265, 79, and 156, respectively].The next component of the detonation model is determining the detonation times at each node contain-ing explosive material. These times are calculated during input processing using a Huygen’s constructionprocedure. Since the detonation times are calculated during input processing, the method is called pro-gram burn. Figure 12 illustrates this procedure in two dimensions.In Figure 12, the detonation is initiated in the node at the lower right cell labeled with a D. Detonationtimes have been calculated for the explosive material in the red cells and have not been calculated for theexplosive material in the green cells. In this figure, the procedure started at the lower left and proceededtoward the right. After sweeping a row, the next row above is swept starting from the left. The procedureis currently in the green cell labeled with an X. While the idea of the construction is given by thisdiscussion of Figure 12, the actual details differ somewhat during EMU execution since the nodes arenot so well ordered in rows and columns as shown in this figure, and a node containing an explosivematerial may be owned by one processor and needed by another processor in the parallel implementation.The detonation times are initially set to a large number. The construction proceeds by sweepingthrough the grid and examining at a given node X the detonation times of the nodes in a spherical neighborhood of node X. If the node spacing is uniform with value 1q, then the radius of the sphericalneighborhood is only slightly greater than√31q.