From f1490637ba0c90ba3229c1f0d0b653be29512f66 Mon Sep 17 00:00:00 2001 From: Liwei Ji Date: Fri, 27 Dec 2024 15:39:57 -0500 Subject: [PATCH] Add thorn ADMconstraints (#50) * ADMconstraints: add admconstraints.cxx * ADMconstraints: add ADM_vars.wl * ADMconstraints: add ADM_rhs.wl * ADMconstraints: add type_in_rhs.nb * ADMconstraints: nv type_in_rhs.nb outside wl * ADMconstraints: update type_in_rhs.nb * ADMconstraints: update ADM_vars.wl * ADMconstraints: update type_in_rhs.nb * ADMconstraints: update type_in_rhs.nb * ADMconstraints: update ADM_rhs.wl * ADMconstraints: add check_rhs.ipynb * ADMconstraints: update check_rhs.ipynb * ADMconstraints: add ADM_set_constraint.wl * ADMconstraints: add ADM_set_constraint.hxx * ADMconstraints: add interface.ccl * ADMconstraints: rename MtC to MC * ADMconstraints: add param.ccl * ADMconstraints: update admconstraints.cxx * ADMconstraints: update admconstraints.cxx * ADMconstraints: remove prefix ADM for variables * ADMconstraints: update admconstraints.cxx * ADMconstraints: update admconstraints.cxx * ADMconstraints: add make.code.defn * ADMconstraints: add configuration.ccl * ADMconstraints: add schedule.ccl * ADMconstraints: make it compile * ADMconstraints: format ADM_rhs.wl * ADMconstraints: treat DexK separatedly to remove warnings * ADMconstraints: treat dexK separatedly to remove warnings * ADMconstraints: add lapse mask * ADMconstraints: add extra SYNCs * ADMconstraints: fix comments in schedule * ADMconstraints: add par lapse_mask_outer_radius * ADMconstraints: mask out region which is larger than a radius * ADMconstraints: fix type of default value for lapse_mask_outer_radius * ADMconstraints: use or instead of and when apply mask * ADMconstraints: sync HC and MC after calculation --- ADMconstraints/configuration.ccl | 3 + ADMconstraints/interface.ccl | 20 + ADMconstraints/param.ccl | 27 + ADMconstraints/schedule.ccl | 52 + ADMconstraints/src/admconstraints.cxx | 145 ++ ADMconstraints/src/lapsemask.cxx | 29 + ADMconstraints/src/make.code.defn | 9 + ADMconstraints/wolfram/ADM_set_constraint.hxx | 1263 +++++++++++++ ADMconstraints/wolfram/ADM_set_constraint.wl | 97 + ADMconstraints/wolfram/check_rhs.ipynb | 270 +++ ADMconstraints/wolfram/type_in_rhs.nb | 1567 +++++++++++++++++ ADMconstraints/wolfram/wl/ADM_rhs.wl | 41 + ADMconstraints/wolfram/wl/ADM_vars.wl | 86 + 13 files changed, 3609 insertions(+) create mode 100644 ADMconstraints/configuration.ccl create mode 100644 ADMconstraints/interface.ccl create mode 100644 ADMconstraints/param.ccl create mode 100644 ADMconstraints/schedule.ccl create mode 100644 ADMconstraints/src/admconstraints.cxx create mode 100644 ADMconstraints/src/lapsemask.cxx create mode 100644 ADMconstraints/src/make.code.defn create mode 100644 ADMconstraints/wolfram/ADM_set_constraint.hxx create mode 100644 ADMconstraints/wolfram/ADM_set_constraint.wl create mode 100644 ADMconstraints/wolfram/check_rhs.ipynb create mode 100644 ADMconstraints/wolfram/type_in_rhs.nb create mode 100644 ADMconstraints/wolfram/wl/ADM_rhs.wl create mode 100644 ADMconstraints/wolfram/wl/ADM_vars.wl diff --git a/ADMconstraints/configuration.ccl b/ADMconstraints/configuration.ccl new file mode 100644 index 00000000..82a6edea --- /dev/null +++ b/ADMconstraints/configuration.ccl @@ -0,0 +1,3 @@ +# Configuration definitions for thorn ADMconstraints + +REQUIRES Arith Loop Derivs diff --git a/ADMconstraints/interface.ccl b/ADMconstraints/interface.ccl new file mode 100644 index 00000000..5c0f7a69 --- /dev/null +++ b/ADMconstraints/interface.ccl @@ -0,0 +1,20 @@ +# Interface definition for thorn ADMconstraints + +IMPLEMENTS: ADMconstraints + +INHERITS: ADMBaseX TmunuBaseX + +USES INCLUDE HEADER: defs.hxx +USES INCLUDE HEADER: derivs.hxx +USES INCLUDE HEADER: div.hxx +USES INCLUDE HEADER: dual.hxx +USES INCLUDE HEADER: loop_device.hxx +USES INCLUDE HEADER: mat.hxx +USES INCLUDE HEADER: simd.hxx +USES INCLUDE HEADER: sum.hxx +USES INCLUDE HEADER: vec.hxx +USES INCLUDE HEADER: vect.hxx + +CCTK_REAL HC TYPE=gf TAGS='checkpoint="no"' "H" + +CCTK_REAL MC TYPE=gf TAGS='parities={-1 +1 +1 +1 -1 +1 +1 +1 -1} checkpoint="no"' { MCx MCy MCz } "M" diff --git a/ADMconstraints/param.ccl b/ADMconstraints/param.ccl new file mode 100644 index 00000000..a0cc7bd3 --- /dev/null +++ b/ADMconstraints/param.ccl @@ -0,0 +1,27 @@ +# Parameter definitions for thorn ADMconstraints + +BOOLEAN calc_constraints "Calculate constraints" STEERABLE=recover +{ +} yes + +CCTK_INT deriv_order "Order of spatial finite differencing" STEERABLE=never +{ + 2 :: "Second order finite difference" + 4 :: "Fourth order finite difference" + 6 :: "Sixth order finite difference" + 8 :: "Eighth order finite difference" +} 4 + +BOOLEAN use_lapse_mask "Exclude the region when lapse is small" STEERABLE=recover +{ +} yes + +CCTK_REAL lapse_mask_cutoff "Exclude the region when lapse is smaller than the cutoff" STEERABLE=always +{ + *:* :: "" +} 0.3 + +CCTK_REAL lapse_mask_outer_radius "Exclude the region when the distance to the origin is larger than the radius" STEERABLE=always +{ + *:* :: "" +} 30.0 diff --git a/ADMconstraints/schedule.ccl b/ADMconstraints/schedule.ccl new file mode 100644 index 00000000..84a550d4 --- /dev/null +++ b/ADMconstraints/schedule.ccl @@ -0,0 +1,52 @@ +# Schedule definitions for thorn ADMconstraints + +STORAGE: HC +STORAGE: MC + + + + +SCHEDULE GROUP ADMconstraints_AnalysisGroup AT analysis +{ +} "Analyse ADM variables" + + +if (calc_constraints) { + SCHEDULE ADMconstraints_Sync IN ADMconstraints_AnalysisGroup + { + LANG: C + OPTIONS: global + SYNC: ADMBaseX::metric + SYNC: ADMBaseX::curv + SYNC: ADMBaseX::lapse + SYNC: ADMBaseX::shift + } "Synchronize" + + SCHEDULE ADMconstraints_CalcConstraints IN ADMconstraints_AnalysisGroup AFTER ADMconstraints_Sync + { + LANG: C + READS: ADMBaseX::metric(everywhere) + READS: ADMBaseX::curv(everywhere) + READS: ADMBaseX::lapse(everywhere) + READS: ADMBaseX::shift(everywhere) + READS: TmunuBaseX::eTtt(interior) + READS: TmunuBaseX::eTti(interior) + READS: TmunuBaseX::eTij(interior) + WRITES: HC(interior) + WRITES: MC(interior) + SYNC: HC MC + } "Calculate ADM Constraints" + + if (use_lapse_mask) { + SCHEDULE ADMconstraints_LapseMask IN ADMconstraints_AnalysisGroup AFTER ADMconstraints_CalcConstraints + { + LANG: C + READS: ADMBaseX::lapse(interior) + READS: HC(interior) + READS: MC(interior) + WRITES: HC(interior) + WRITES: MC(interior) + SYNC: HC MC + } "Apply lapse mask" + } +} diff --git a/ADMconstraints/src/admconstraints.cxx b/ADMconstraints/src/admconstraints.cxx new file mode 100644 index 00000000..18abbc2f --- /dev/null +++ b/ADMconstraints/src/admconstraints.cxx @@ -0,0 +1,145 @@ +#include +#include +#include + +#ifdef __CUDACC__ +// Disable CCTK_DEBUG since the debug information takes too much +// parameter space to launch the kernels +#ifdef CCTK_DEBUG +#undef CCTK_DEBUG +#endif +#endif + +#include +#include +#include + +#ifdef __CUDACC__ +#include +#endif + +#include + +namespace ADMconstraints { +using namespace Arith; +using namespace Loop; + +template +CCTK_DEVICE CCTK_HOST CCTK_ATTRIBUTE_ALWAYS_INLINE inline T Power(T x, int y) { + return (y == 2) ? Arith::pow2(x) : Arith::pown(x, y); +} + +extern "C" void ADMconstraints_Sync(CCTK_ARGUMENTS) { + // do nothing +} + +extern "C" void ADMconstraints_CalcConstraints(CCTK_ARGUMENTS) { + DECLARE_CCTK_ARGUMENTSX_ADMconstraints_CalcConstraints; + DECLARE_CCTK_PARAMETERS; + + for (int d = 0; d < 3; ++d) + if (cctk_nghostzones[d] < deriv_order / 2) + CCTK_VERROR("Need at least %d ghost zones", deriv_order / 2); + + const vect dx{ + CCTK_DELTA_SPACE(0), + CCTK_DELTA_SPACE(1), + CCTK_DELTA_SPACE(2), + }; + + const array indextype = {0, 0, 0}; + const array nghostzones = {cctk_nghostzones[0], cctk_nghostzones[1], + cctk_nghostzones[2]}; + vect imin, imax; + GridDescBase(cctkGH).box_int<0, 0, 0>(nghostzones, imin, imax); + // suffix 2: with ghost zones, suffix 5: without ghost zones + const GF3D2layout layout2(cctkGH, indextype); + const GF3D5layout layout5(imin, imax); + + // Input grid functions + const smat, 3> gf_gam{gxx, gxy, gxz, gyy, gyz, gzz}; + const smat, 3> gf_exK{kxx, kxy, kxz, kyy, kyz, kzz}; + const GF3D2 &gf_alpha = alp; + const vec, 3> gf_beta{betax, betay, betaz}; + + // More input grid functions + const GF3D2 &gf_eTtt = eTtt; + const vec, 3> gf_eTt{eTtx, eTty, eTtz}; + const smat, 3> gf_eT{eTxx, eTxy, eTxz, + eTyy, eTyz, eTzz}; + + // Output grid functions + const GF3D2 &gf_HC = HC; + const vec, 3> gf_MC{MCx, MCy, MCz}; + + // Define derivs lambdas + const auto calccopy = [&](const auto &gf, const auto &gf0) { + Derivs::calc_copy<0, 0, 0>(gf, layout5, grid, gf0); + }; + const auto calcderivs = [&](const auto &gf, const auto &dgf, + const auto &gf0) { + Derivs::calc_derivs<0, 0, 0>(gf, dgf, layout5, grid, gf0, dx, deriv_order); + }; + const auto calcderivs2 = [&](const auto &gf, const auto &dgf, + const auto &ddgf, const auto &gf0) { + Derivs::calc_derivs2<0, 0, 0>(gf, dgf, ddgf, layout5, grid, gf0, dx, + deriv_order); + }; + + // Tile variables for derivatives and so on + const int ntmps = 88; + GF3D5vector tmps(layout5, ntmps); + int itmp = 0; + + const auto make_gf = [&]() { return GF3D5(tmps(itmp++)); }; + const auto make_vec = [&](const auto &f) { + return vec, 3>([&](int) { return f(); }); + }; + const auto make_mat = [&](const auto &f) { + return smat, 3>([&](int, int) { return f(); }); + }; + const auto make_vec_gf = [&]() { return make_vec(make_gf); }; + const auto make_mat_gf = [&]() { return make_mat(make_gf); }; + const auto make_mat_vec_gf = [&]() { return make_mat(make_vec_gf); }; + const auto make_mat_mat_gf = [&]() { return make_mat(make_mat_gf); }; + + const smat, 3> tl_gam(make_mat_gf()); + const smat, 3>, 3> tl_dgam(make_mat_vec_gf()); + const smat, 3>, 3> tl_ddgam(make_mat_mat_gf()); + calcderivs2(tl_gam, tl_dgam, tl_ddgam, gf_gam); + + const smat, 3> tl_exK(make_mat_gf()); + const smat, 3>, 3> tl_dexK(make_mat_vec_gf()); + calcderivs(tl_exK, tl_dexK, gf_exK); + + const GF3D5 tl_alpha(make_gf()); + calccopy(tl_alpha, gf_alpha); + + const vec, 3> tl_beta(make_vec_gf()); + calccopy(tl_beta, gf_beta); + + if (itmp != ntmps) + CCTK_VERROR("Wrong number of temporary variables: ntmps=%d itmp=%d", ntmps, + itmp); + itmp = -1; + + // simd types + typedef simd vreal; + typedef simdl vbool; + constexpr size_t vsize = tuple_size_v; + + // parameters + const CCTK_REAL cpi = M_PI; + +#ifdef __CUDACC__ + const nvtxRangeId_t range = nvtxRangeStartA("ADMconstraints::constraints"); +#endif + +#include "../wolfram/ADM_set_constraint.hxx" + +#ifdef __CUDACC__ + nvtxRangeEnd(range); +#endif +} + +} // namespace ADMconstraints diff --git a/ADMconstraints/src/lapsemask.cxx b/ADMconstraints/src/lapsemask.cxx new file mode 100644 index 00000000..a344c887 --- /dev/null +++ b/ADMconstraints/src/lapsemask.cxx @@ -0,0 +1,29 @@ +#include + +#include +#include + +namespace ADMconstraints { +using namespace Arith; +using namespace Loop; + +extern "C" void ADMconstraints_LapseMask(CCTK_ARGUMENTS) { + DECLARE_CCTK_ARGUMENTSX_ADMconstraints_LapseMask; + DECLARE_CCTK_PARAMETERS; + + const CCTK_REAL local_cutoff = lapse_mask_cutoff; + const CCTK_REAL local_outer_radius = lapse_mask_outer_radius; + + grid.loop_int_device<0, 0, 0>( + grid.nghostzones, [=] ARITH_DEVICE(const PointDesc &p) ARITH_INLINE { + const CCTK_REAL rad = sqrt(p.x * p.x + p.y * p.y + p.z * p.z); + if (alp(p.I) < local_cutoff || rad > local_outer_radius) { + HC(p.I) = 0.0; + MCx(p.I) = 0.0; + MCy(p.I) = 0.0; + MCz(p.I) = 0.0; + } + }); +} + +} // namespace ADMconstraints diff --git a/ADMconstraints/src/make.code.defn b/ADMconstraints/src/make.code.defn new file mode 100644 index 00000000..c027665f --- /dev/null +++ b/ADMconstraints/src/make.code.defn @@ -0,0 +1,9 @@ +# Main make.code.defn file for thorn Z4c + +# Source files in this directory +SRCS = \ + admconstraints.cxx \ + lapsemask.cxx + +# Subdirectories containing source files +SUBDIRS = diff --git a/ADMconstraints/wolfram/ADM_set_constraint.hxx b/ADMconstraints/wolfram/ADM_set_constraint.hxx new file mode 100644 index 00000000..57514b25 --- /dev/null +++ b/ADMconstraints/wolfram/ADM_set_constraint.hxx @@ -0,0 +1,1263 @@ +/* ADM_set_constraint.hxx */ +/* Produced with Mathematica */ + +#ifndef ADM_SET_CONSTRAINT_HXX +#define ADM_SET_CONSTRAINT_HXX + +const GF3D2 &local_HC = gf_HC; +const GF3D2 &local_MC1 = gf_MC(0); +const GF3D2 &local_MC2 = gf_MC(1); +const GF3D2 &local_MC3 = gf_MC(2); + +noinline([&]() __attribute__((__flatten__, __hot__)) { + grid.loop_int_device<0, 0, 0, vsize>( + grid.nghostzones, [=] ARITH_DEVICE(const PointDesc &p) ARITH_INLINE { + const vbool mask = mask_for_loop_tail(p.i, p.imax); + const GF3D2index index2(layout2, p.I); + const GF3D5index index5(layout5, p.I); + +const auto &tmp_eTtt = gf_eTtt(mask, index2); +const auto &tmp_eTt = gf_eTt(mask, index2); +const auto &tmp_eT = gf_eT(mask, index2); +const auto &tmp_gam = tl_gam(mask, index5); +const auto &tmp_exK = tl_exK(mask, index5); +const auto &tmp_alpha = tl_alpha(mask, index5); +const auto &tmp_beta = tl_beta(mask, index5); +const auto &tmp_dgam = tl_dgam(mask, index5); +const auto &tmp_dexK = tl_dexK(mask, index5); +const auto &tmp_ddgam = tl_ddgam(mask, index5); + +const vreal eTtt = tmp_eTtt; +const vreal eTt1 = tmp_eTt(0); +const vreal eTt2 = tmp_eTt(1); +const vreal eTt3 = tmp_eTt(2); +const vreal eT11 = tmp_eT(0,0); +const vreal eT12 = tmp_eT(0,1); +const vreal eT13 = tmp_eT(0,2); +const vreal eT22 = tmp_eT(1,1); +const vreal eT23 = tmp_eT(1,2); +const vreal eT33 = tmp_eT(2,2); +const vreal gam11 = tmp_gam(0,0); +const vreal gam12 = tmp_gam(0,1); +const vreal gam13 = tmp_gam(0,2); +const vreal gam22 = tmp_gam(1,1); +const vreal gam23 = tmp_gam(1,2); +const vreal gam33 = tmp_gam(2,2); +const vreal exK11 = tmp_exK(0,0); +const vreal exK12 = tmp_exK(0,1); +const vreal exK13 = tmp_exK(0,2); +const vreal exK22 = tmp_exK(1,1); +const vreal exK23 = tmp_exK(1,2); +const vreal exK33 = tmp_exK(2,2); +const vreal alpha = tmp_alpha; +const vreal beta1 = tmp_beta(0); +const vreal beta2 = tmp_beta(1); +const vreal beta3 = tmp_beta(2); +const vreal dgam111 = tmp_dgam(0,0)(0); +const vreal dgam112 = tmp_dgam(0,1)(0); +const vreal dgam113 = tmp_dgam(0,2)(0); +const vreal dgam122 = tmp_dgam(1,1)(0); +const vreal dgam123 = tmp_dgam(1,2)(0); +const vreal dgam133 = tmp_dgam(2,2)(0); +const vreal dgam211 = tmp_dgam(0,0)(1); +const vreal dgam212 = tmp_dgam(0,1)(1); +const vreal dgam213 = tmp_dgam(0,2)(1); +const vreal dgam222 = tmp_dgam(1,1)(1); +const vreal dgam223 = tmp_dgam(1,2)(1); +const vreal dgam233 = tmp_dgam(2,2)(1); +const vreal dgam311 = tmp_dgam(0,0)(2); +const vreal dgam312 = tmp_dgam(0,1)(2); +const vreal dgam313 = tmp_dgam(0,2)(2); +const vreal dgam322 = tmp_dgam(1,1)(2); +const vreal dgam323 = tmp_dgam(1,2)(2); +const vreal dgam333 = tmp_dgam(2,2)(2); +const vreal dexK112 = tmp_dexK(0,1)(0); +const vreal dexK113 = tmp_dexK(0,2)(0); +const vreal dexK122 = tmp_dexK(1,1)(0); +const vreal dexK123 = tmp_dexK(1,2)(0); +const vreal dexK133 = tmp_dexK(2,2)(0); +const vreal dexK211 = tmp_dexK(0,0)(1); +const vreal dexK212 = tmp_dexK(0,1)(1); +const vreal dexK213 = tmp_dexK(0,2)(1); +const vreal dexK223 = tmp_dexK(1,2)(1); +const vreal dexK233 = tmp_dexK(2,2)(1); +const vreal dexK311 = tmp_dexK(0,0)(2); +const vreal dexK312 = tmp_dexK(0,1)(2); +const vreal dexK313 = tmp_dexK(0,2)(2); +const vreal dexK322 = tmp_dexK(1,1)(2); +const vreal dexK323 = tmp_dexK(1,2)(2); +const vreal ddgam1111 = tmp_ddgam(0,0)(0,0); +const vreal ddgam1112 = tmp_ddgam(0,1)(0,0); +const vreal ddgam1113 = tmp_ddgam(0,2)(0,0); +const vreal ddgam1122 = tmp_ddgam(1,1)(0,0); +const vreal ddgam1123 = tmp_ddgam(1,2)(0,0); +const vreal ddgam1133 = tmp_ddgam(2,2)(0,0); +const vreal ddgam1211 = tmp_ddgam(0,0)(0,1); +const vreal ddgam1212 = tmp_ddgam(0,1)(0,1); +const vreal ddgam1213 = tmp_ddgam(0,2)(0,1); +const vreal ddgam1222 = tmp_ddgam(1,1)(0,1); +const vreal ddgam1223 = tmp_ddgam(1,2)(0,1); +const vreal ddgam1233 = tmp_ddgam(2,2)(0,1); +const vreal ddgam1311 = tmp_ddgam(0,0)(0,2); +const vreal ddgam1312 = tmp_ddgam(0,1)(0,2); +const vreal ddgam1313 = tmp_ddgam(0,2)(0,2); +const vreal ddgam1322 = tmp_ddgam(1,1)(0,2); +const vreal ddgam1323 = tmp_ddgam(1,2)(0,2); +const vreal ddgam1333 = tmp_ddgam(2,2)(0,2); +const vreal ddgam2211 = tmp_ddgam(0,0)(1,1); +const vreal ddgam2212 = tmp_ddgam(0,1)(1,1); +const vreal ddgam2213 = tmp_ddgam(0,2)(1,1); +const vreal ddgam2222 = tmp_ddgam(1,1)(1,1); +const vreal ddgam2223 = tmp_ddgam(1,2)(1,1); +const vreal ddgam2233 = tmp_ddgam(2,2)(1,1); +const vreal ddgam2311 = tmp_ddgam(0,0)(1,2); +const vreal ddgam2312 = tmp_ddgam(0,1)(1,2); +const vreal ddgam2313 = tmp_ddgam(0,2)(1,2); +const vreal ddgam2322 = tmp_ddgam(1,1)(1,2); +const vreal ddgam2323 = tmp_ddgam(1,2)(1,2); +const vreal ddgam2333 = tmp_ddgam(2,2)(1,2); +const vreal ddgam3311 = tmp_ddgam(0,0)(2,2); +const vreal ddgam3312 = tmp_ddgam(0,1)(2,2); +const vreal ddgam3313 = tmp_ddgam(0,2)(2,2); +const vreal ddgam3322 = tmp_ddgam(1,1)(2,2); +const vreal ddgam3323 = tmp_ddgam(1,2)(2,2); +const vreal ddgam3333 = tmp_ddgam(2,2)(2,2); + +vreal detinvgam += +1/(-(Power(gam13,2)*gam22) + 2*gam12*gam13*gam23 - gam11*Power(gam23,2) - + Power(gam12,2)*gam33 + gam11*gam22*gam33) +; + +vreal invgam11 += +detinvgam*(-Power(gam23,2) + gam22*gam33) +; + +vreal invgam12 += +detinvgam*(gam13*gam23 - gam12*gam33) +; + +vreal invgam13 += +detinvgam*(-(gam13*gam22) + gam12*gam23) +; + +vreal invgam22 += +detinvgam*(-Power(gam13,2) + gam11*gam33) +; + +vreal invgam23 += +detinvgam*(gam12*gam13 - gam11*gam23) +; + +vreal invgam33 += +detinvgam*(-Power(gam12,2) + gam11*gam22) +; + +vreal GamDDD111 += +dgam111/2. +; + +vreal GamDDD112 += +dgam211/2. +; + +vreal GamDDD113 += +dgam311/2. +; + +vreal GamDDD122 += +-0.5*dgam122 + dgam212 +; + +vreal GamDDD123 += +(-dgam123 + dgam213 + dgam312)/2. +; + +vreal GamDDD133 += +-0.5*dgam133 + dgam313 +; + +vreal GamDDD211 += +dgam112 - dgam211/2. +; + +vreal GamDDD212 += +dgam122/2. +; + +vreal GamDDD213 += +(dgam123 - dgam213 + dgam312)/2. +; + +vreal GamDDD222 += +dgam222/2. +; + +vreal GamDDD223 += +dgam322/2. +; + +vreal GamDDD233 += +-0.5*dgam233 + dgam323 +; + +vreal GamDDD311 += +dgam113 - dgam311/2. +; + +vreal GamDDD312 += +(dgam123 + dgam213 - dgam312)/2. +; + +vreal GamDDD313 += +dgam133/2. +; + +vreal GamDDD322 += +dgam223 - dgam322/2. +; + +vreal GamDDD323 += +dgam233/2. +; + +vreal GamDDD333 += +dgam333/2. +; + +vreal Gam111 += +GamDDD111*invgam11 + GamDDD211*invgam12 + GamDDD311*invgam13 +; + +vreal Gam112 += +GamDDD112*invgam11 + GamDDD212*invgam12 + GamDDD312*invgam13 +; + +vreal Gam113 += +GamDDD113*invgam11 + GamDDD213*invgam12 + GamDDD313*invgam13 +; + +vreal Gam122 += +GamDDD122*invgam11 + GamDDD222*invgam12 + GamDDD322*invgam13 +; + +vreal Gam123 += +GamDDD123*invgam11 + GamDDD223*invgam12 + GamDDD323*invgam13 +; + +vreal Gam133 += +GamDDD133*invgam11 + GamDDD233*invgam12 + GamDDD333*invgam13 +; + +vreal Gam211 += +GamDDD111*invgam12 + GamDDD211*invgam22 + GamDDD311*invgam23 +; + +vreal Gam212 += +GamDDD112*invgam12 + GamDDD212*invgam22 + GamDDD312*invgam23 +; + +vreal Gam213 += +GamDDD113*invgam12 + GamDDD213*invgam22 + GamDDD313*invgam23 +; + +vreal Gam222 += +GamDDD122*invgam12 + GamDDD222*invgam22 + GamDDD322*invgam23 +; + +vreal Gam223 += +GamDDD123*invgam12 + GamDDD223*invgam22 + GamDDD323*invgam23 +; + +vreal Gam233 += +GamDDD133*invgam12 + GamDDD233*invgam22 + GamDDD333*invgam23 +; + +vreal Gam311 += +GamDDD111*invgam13 + GamDDD211*invgam23 + GamDDD311*invgam33 +; + +vreal Gam312 += +GamDDD112*invgam13 + GamDDD212*invgam23 + GamDDD312*invgam33 +; + +vreal Gam313 += +GamDDD113*invgam13 + GamDDD213*invgam23 + GamDDD313*invgam33 +; + +vreal Gam322 += +GamDDD122*invgam13 + GamDDD222*invgam23 + GamDDD322*invgam33 +; + +vreal Gam323 += +GamDDD123*invgam13 + GamDDD223*invgam23 + GamDDD323*invgam33 +; + +vreal Gam333 += +GamDDD133*invgam13 + GamDDD233*invgam23 + GamDDD333*invgam33 +; + +vreal tr1dGam11 += +(ddgam1111*invgam11)/2. + ddgam1112*invgam12 - + 2*dgam112*GamDDD111*invgam11*invgam12 - + dgam211*GamDDD111*invgam11*invgam12 - + dgam122*GamDDD111*Power(invgam12,2) - + dgam212*GamDDD111*Power(invgam12,2) - + dgam112*GamDDD211*Power(invgam12,2) - + dgam211*GamDDD211*Power(invgam12,2) + ddgam1113*invgam13 - + 2*dgam113*GamDDD111*invgam11*invgam13 - + dgam311*GamDDD111*invgam11*invgam13 - + 2*dgam123*GamDDD111*invgam12*invgam13 - + dgam213*GamDDD111*invgam12*invgam13 - + dgam312*GamDDD111*invgam12*invgam13 - + dgam113*GamDDD211*invgam12*invgam13 - + dgam311*GamDDD211*invgam12*invgam13 - + dgam112*GamDDD311*invgam12*invgam13 - + dgam211*GamDDD311*invgam12*invgam13 - + dgam133*GamDDD111*Power(invgam13,2) - + dgam313*GamDDD111*Power(invgam13,2) - + dgam113*GamDDD311*Power(invgam13,2) - + dgam311*GamDDD311*Power(invgam13,2) - + dgam111*invgam11*(GamDDD111*invgam11 + GamDDD211*invgam12 + + GamDDD311*invgam13) + ddgam1212*invgam22 - (ddgam2211*invgam22)/2. - + dgam212*GamDDD111*invgam11*invgam22 - + dgam112*GamDDD211*invgam11*invgam22 - + dgam222*GamDDD111*invgam12*invgam22 - + dgam122*GamDDD211*invgam12*invgam22 - + 2*dgam212*GamDDD211*invgam12*invgam22 - + dgam223*GamDDD111*invgam13*invgam22 - + dgam123*GamDDD211*invgam13*invgam22 - + dgam312*GamDDD211*invgam13*invgam22 - + dgam212*GamDDD311*invgam13*invgam22 - + dgam222*GamDDD211*Power(invgam22,2) + ddgam1213*invgam23 + + ddgam1312*invgam23 - ddgam2311*invgam23 - + dgam213*GamDDD111*invgam11*invgam23 - + dgam312*GamDDD111*invgam11*invgam23 - + dgam113*GamDDD211*invgam11*invgam23 - + dgam112*GamDDD311*invgam11*invgam23 - + dgam223*GamDDD111*invgam12*invgam23 - + dgam322*GamDDD111*invgam12*invgam23 - + dgam123*GamDDD211*invgam12*invgam23 - + 2*dgam213*GamDDD211*invgam12*invgam23 - + dgam312*GamDDD211*invgam12*invgam23 - + dgam122*GamDDD311*invgam12*invgam23 - + dgam212*GamDDD311*invgam12*invgam23 - + dgam233*GamDDD111*invgam13*invgam23 - + dgam323*GamDDD111*invgam13*invgam23 - + dgam133*GamDDD211*invgam13*invgam23 - + dgam313*GamDDD211*invgam13*invgam23 - + dgam123*GamDDD311*invgam13*invgam23 - + dgam213*GamDDD311*invgam13*invgam23 - + 2*dgam312*GamDDD311*invgam13*invgam23 - + 2*dgam223*GamDDD211*invgam22*invgam23 - + dgam322*GamDDD211*invgam22*invgam23 - + dgam222*GamDDD311*invgam22*invgam23 - + dgam233*GamDDD211*Power(invgam23,2) - + dgam323*GamDDD211*Power(invgam23,2) - + dgam223*GamDDD311*Power(invgam23,2) - + dgam322*GamDDD311*Power(invgam23,2) + ddgam1313*invgam33 - + (ddgam3311*invgam33)/2. - dgam313*GamDDD111*invgam11*invgam33 - + dgam113*GamDDD311*invgam11*invgam33 - + dgam323*GamDDD111*invgam12*invgam33 - + dgam313*GamDDD211*invgam12*invgam33 - + dgam123*GamDDD311*invgam12*invgam33 - + dgam213*GamDDD311*invgam12*invgam33 - + dgam333*GamDDD111*invgam13*invgam33 - + dgam133*GamDDD311*invgam13*invgam33 - + 2*dgam313*GamDDD311*invgam13*invgam33 - + dgam323*GamDDD211*invgam22*invgam33 - + dgam223*GamDDD311*invgam22*invgam33 - + dgam333*GamDDD211*invgam23*invgam33 - + dgam233*GamDDD311*invgam23*invgam33 - + 2*dgam323*GamDDD311*invgam23*invgam33 - dgam333*GamDDD311*Power(invgam33,2) +; + +vreal tr1dGam12 += +(ddgam1211*invgam11 + ddgam1122*invgam12 + ddgam2211*invgam12 - + 4*dgam112*GamDDD112*invgam11*invgam12 - + 2*dgam211*GamDDD112*invgam11*invgam12 - + 2*dgam122*GamDDD112*Power(invgam12,2) - + 2*dgam212*GamDDD112*Power(invgam12,2) - + 2*dgam112*GamDDD212*Power(invgam12,2) - + 2*dgam211*GamDDD212*Power(invgam12,2) + ddgam1123*invgam13 + + ddgam1213*invgam13 - ddgam1312*invgam13 + ddgam2311*invgam13 - + 4*dgam113*GamDDD112*invgam11*invgam13 - + 2*dgam311*GamDDD112*invgam11*invgam13 - + 4*dgam123*GamDDD112*invgam12*invgam13 - + 2*dgam213*GamDDD112*invgam12*invgam13 - + 2*dgam312*GamDDD112*invgam12*invgam13 - + 2*dgam113*GamDDD212*invgam12*invgam13 - + 2*dgam311*GamDDD212*invgam12*invgam13 - + 2*dgam112*GamDDD312*invgam12*invgam13 - + 2*dgam211*GamDDD312*invgam12*invgam13 - + 2*dgam133*GamDDD112*Power(invgam13,2) - + 2*dgam313*GamDDD112*Power(invgam13,2) - + 2*dgam113*GamDDD312*Power(invgam13,2) - + 2*dgam311*GamDDD312*Power(invgam13,2) - + 2*dgam111*invgam11*(GamDDD112*invgam11 + GamDDD212*invgam12 + + GamDDD312*invgam13) + ddgam1222*invgam22 - + 2*dgam212*GamDDD112*invgam11*invgam22 - + 2*dgam112*GamDDD212*invgam11*invgam22 - + 2*dgam222*GamDDD112*invgam12*invgam22 - + 2*dgam122*GamDDD212*invgam12*invgam22 - + 4*dgam212*GamDDD212*invgam12*invgam22 - + 2*dgam223*GamDDD112*invgam13*invgam22 - + 2*dgam123*GamDDD212*invgam13*invgam22 - + 2*dgam312*GamDDD212*invgam13*invgam22 - + 2*dgam212*GamDDD312*invgam13*invgam22 - + 2*dgam222*GamDDD212*Power(invgam22,2) + ddgam1223*invgam23 + + ddgam1322*invgam23 + ddgam2213*invgam23 - ddgam2312*invgam23 - + 2*dgam213*GamDDD112*invgam11*invgam23 - + 2*dgam312*GamDDD112*invgam11*invgam23 - + 2*dgam113*GamDDD212*invgam11*invgam23 - + 2*dgam112*GamDDD312*invgam11*invgam23 - + 2*dgam223*GamDDD112*invgam12*invgam23 - + 2*dgam322*GamDDD112*invgam12*invgam23 - + 2*dgam123*GamDDD212*invgam12*invgam23 - + 4*dgam213*GamDDD212*invgam12*invgam23 - + 2*dgam312*GamDDD212*invgam12*invgam23 - + 2*dgam122*GamDDD312*invgam12*invgam23 - + 2*dgam212*GamDDD312*invgam12*invgam23 - + 2*dgam233*GamDDD112*invgam13*invgam23 - + 2*dgam323*GamDDD112*invgam13*invgam23 - + 2*dgam133*GamDDD212*invgam13*invgam23 - + 2*dgam313*GamDDD212*invgam13*invgam23 - + 2*dgam123*GamDDD312*invgam13*invgam23 - + 2*dgam213*GamDDD312*invgam13*invgam23 - + 4*dgam312*GamDDD312*invgam13*invgam23 - + 4*dgam223*GamDDD212*invgam22*invgam23 - + 2*dgam322*GamDDD212*invgam22*invgam23 - + 2*dgam222*GamDDD312*invgam22*invgam23 - + 2*dgam233*GamDDD212*Power(invgam23,2) - + 2*dgam323*GamDDD212*Power(invgam23,2) - + 2*dgam223*GamDDD312*Power(invgam23,2) - + 2*dgam322*GamDDD312*Power(invgam23,2) + ddgam1323*invgam33 + + ddgam2313*invgam33 - ddgam3312*invgam33 - + 2*dgam313*GamDDD112*invgam11*invgam33 - + 2*dgam113*GamDDD312*invgam11*invgam33 - + 2*dgam323*GamDDD112*invgam12*invgam33 - + 2*dgam313*GamDDD212*invgam12*invgam33 - + 2*dgam123*GamDDD312*invgam12*invgam33 - + 2*dgam213*GamDDD312*invgam12*invgam33 - + 2*dgam333*GamDDD112*invgam13*invgam33 - + 2*dgam133*GamDDD312*invgam13*invgam33 - + 4*dgam313*GamDDD312*invgam13*invgam33 - + 2*dgam323*GamDDD212*invgam22*invgam33 - + 2*dgam223*GamDDD312*invgam22*invgam33 - + 2*dgam333*GamDDD212*invgam23*invgam33 - + 2*dgam233*GamDDD312*invgam23*invgam33 - + 4*dgam323*GamDDD312*invgam23*invgam33 - + 2*dgam333*GamDDD312*Power(invgam33,2))/2. +; + +vreal tr1dGam13 += +(ddgam1311*invgam11 + ddgam1123*invgam12 - ddgam1213*invgam12 + + ddgam1312*invgam12 + ddgam2311*invgam12 - + 4*dgam112*GamDDD113*invgam11*invgam12 - + 2*dgam211*GamDDD113*invgam11*invgam12 - + 2*dgam122*GamDDD113*Power(invgam12,2) - + 2*dgam212*GamDDD113*Power(invgam12,2) - + 2*dgam112*GamDDD213*Power(invgam12,2) - + 2*dgam211*GamDDD213*Power(invgam12,2) + ddgam1133*invgam13 + + ddgam3311*invgam13 - 4*dgam113*GamDDD113*invgam11*invgam13 - + 2*dgam311*GamDDD113*invgam11*invgam13 - + 4*dgam123*GamDDD113*invgam12*invgam13 - + 2*dgam213*GamDDD113*invgam12*invgam13 - + 2*dgam312*GamDDD113*invgam12*invgam13 - + 2*dgam113*GamDDD213*invgam12*invgam13 - + 2*dgam311*GamDDD213*invgam12*invgam13 - + 2*dgam112*GamDDD313*invgam12*invgam13 - + 2*dgam211*GamDDD313*invgam12*invgam13 - + 2*dgam133*GamDDD113*Power(invgam13,2) - + 2*dgam313*GamDDD113*Power(invgam13,2) - + 2*dgam113*GamDDD313*Power(invgam13,2) - + 2*dgam311*GamDDD313*Power(invgam13,2) - + 2*dgam111*invgam11*(GamDDD113*invgam11 + GamDDD213*invgam12 + + GamDDD313*invgam13) + ddgam1223*invgam22 - ddgam2213*invgam22 + + ddgam2312*invgam22 - 2*dgam212*GamDDD113*invgam11*invgam22 - + 2*dgam112*GamDDD213*invgam11*invgam22 - + 2*dgam222*GamDDD113*invgam12*invgam22 - + 2*dgam122*GamDDD213*invgam12*invgam22 - + 4*dgam212*GamDDD213*invgam12*invgam22 - + 2*dgam223*GamDDD113*invgam13*invgam22 - + 2*dgam123*GamDDD213*invgam13*invgam22 - + 2*dgam312*GamDDD213*invgam13*invgam22 - + 2*dgam212*GamDDD313*invgam13*invgam22 - + 2*dgam222*GamDDD213*Power(invgam22,2) + ddgam1233*invgam23 + + ddgam1323*invgam23 - ddgam2313*invgam23 + ddgam3312*invgam23 - + 2*dgam213*GamDDD113*invgam11*invgam23 - + 2*dgam312*GamDDD113*invgam11*invgam23 - + 2*dgam113*GamDDD213*invgam11*invgam23 - + 2*dgam112*GamDDD313*invgam11*invgam23 - + 2*dgam223*GamDDD113*invgam12*invgam23 - + 2*dgam322*GamDDD113*invgam12*invgam23 - + 2*dgam123*GamDDD213*invgam12*invgam23 - + 4*dgam213*GamDDD213*invgam12*invgam23 - + 2*dgam312*GamDDD213*invgam12*invgam23 - + 2*dgam122*GamDDD313*invgam12*invgam23 - + 2*dgam212*GamDDD313*invgam12*invgam23 - + 2*dgam233*GamDDD113*invgam13*invgam23 - + 2*dgam323*GamDDD113*invgam13*invgam23 - + 2*dgam133*GamDDD213*invgam13*invgam23 - + 2*dgam313*GamDDD213*invgam13*invgam23 - + 2*dgam123*GamDDD313*invgam13*invgam23 - + 2*dgam213*GamDDD313*invgam13*invgam23 - + 4*dgam312*GamDDD313*invgam13*invgam23 - + 4*dgam223*GamDDD213*invgam22*invgam23 - + 2*dgam322*GamDDD213*invgam22*invgam23 - + 2*dgam222*GamDDD313*invgam22*invgam23 - + 2*dgam233*GamDDD213*Power(invgam23,2) - + 2*dgam323*GamDDD213*Power(invgam23,2) - + 2*dgam223*GamDDD313*Power(invgam23,2) - + 2*dgam322*GamDDD313*Power(invgam23,2) + ddgam1333*invgam33 - + 2*dgam313*GamDDD113*invgam11*invgam33 - + 2*dgam113*GamDDD313*invgam11*invgam33 - + 2*dgam323*GamDDD113*invgam12*invgam33 - + 2*dgam313*GamDDD213*invgam12*invgam33 - + 2*dgam123*GamDDD313*invgam12*invgam33 - + 2*dgam213*GamDDD313*invgam12*invgam33 - + 2*dgam333*GamDDD113*invgam13*invgam33 - + 2*dgam133*GamDDD313*invgam13*invgam33 - + 4*dgam313*GamDDD313*invgam13*invgam33 - + 2*dgam323*GamDDD213*invgam22*invgam33 - + 2*dgam223*GamDDD313*invgam22*invgam33 - + 2*dgam333*GamDDD213*invgam23*invgam33 - + 2*dgam233*GamDDD313*invgam23*invgam33 - + 4*dgam323*GamDDD313*invgam23*invgam33 - + 2*dgam333*GamDDD313*Power(invgam33,2))/2. +; + +vreal tr1dGam22 += +-0.5*(ddgam1122*invgam11) + ddgam1212*invgam11 - + dgam111*GamDDD122*Power(invgam11,2) + ddgam2212*invgam12 - + 2*dgam112*GamDDD122*invgam11*invgam12 - + dgam211*GamDDD122*invgam11*invgam12 - + dgam111*GamDDD222*invgam11*invgam12 - + dgam122*GamDDD122*Power(invgam12,2) - + dgam212*GamDDD122*Power(invgam12,2) - + dgam112*GamDDD222*Power(invgam12,2) - + dgam211*GamDDD222*Power(invgam12,2) + ddgam1223*invgam13 - + ddgam1322*invgam13 + ddgam2312*invgam13 - + 2*dgam113*GamDDD122*invgam11*invgam13 - + dgam311*GamDDD122*invgam11*invgam13 - + dgam111*GamDDD322*invgam11*invgam13 - + 2*dgam123*GamDDD122*invgam12*invgam13 - + dgam213*GamDDD122*invgam12*invgam13 - + dgam312*GamDDD122*invgam12*invgam13 - + dgam113*GamDDD222*invgam12*invgam13 - + dgam311*GamDDD222*invgam12*invgam13 - + dgam112*GamDDD322*invgam12*invgam13 - + dgam211*GamDDD322*invgam12*invgam13 - + dgam133*GamDDD122*Power(invgam13,2) - + dgam313*GamDDD122*Power(invgam13,2) - + dgam113*GamDDD322*Power(invgam13,2) - + dgam311*GamDDD322*Power(invgam13,2) + (ddgam2222*invgam22)/2. - + dgam212*GamDDD122*invgam11*invgam22 - + dgam112*GamDDD222*invgam11*invgam22 - + dgam222*GamDDD122*invgam12*invgam22 - + dgam122*GamDDD222*invgam12*invgam22 - + 2*dgam212*GamDDD222*invgam12*invgam22 - + dgam223*GamDDD122*invgam13*invgam22 - + dgam123*GamDDD222*invgam13*invgam22 - + dgam312*GamDDD222*invgam13*invgam22 - + dgam212*GamDDD322*invgam13*invgam22 - + dgam222*GamDDD222*Power(invgam22,2) + ddgam2223*invgam23 - + dgam213*GamDDD122*invgam11*invgam23 - + dgam312*GamDDD122*invgam11*invgam23 - + dgam113*GamDDD222*invgam11*invgam23 - + dgam112*GamDDD322*invgam11*invgam23 - + dgam223*GamDDD122*invgam12*invgam23 - + dgam322*GamDDD122*invgam12*invgam23 - + dgam123*GamDDD222*invgam12*invgam23 - + 2*dgam213*GamDDD222*invgam12*invgam23 - + dgam312*GamDDD222*invgam12*invgam23 - + dgam122*GamDDD322*invgam12*invgam23 - + dgam212*GamDDD322*invgam12*invgam23 - + dgam233*GamDDD122*invgam13*invgam23 - + dgam323*GamDDD122*invgam13*invgam23 - + dgam133*GamDDD222*invgam13*invgam23 - + dgam313*GamDDD222*invgam13*invgam23 - + dgam123*GamDDD322*invgam13*invgam23 - + dgam213*GamDDD322*invgam13*invgam23 - + 2*dgam312*GamDDD322*invgam13*invgam23 - + 2*dgam223*GamDDD222*invgam22*invgam23 - + dgam322*GamDDD222*invgam22*invgam23 - + dgam222*GamDDD322*invgam22*invgam23 - + dgam233*GamDDD222*Power(invgam23,2) - + dgam323*GamDDD222*Power(invgam23,2) - + dgam223*GamDDD322*Power(invgam23,2) - + dgam322*GamDDD322*Power(invgam23,2) + ddgam2323*invgam33 - + (ddgam3322*invgam33)/2. - dgam313*GamDDD122*invgam11*invgam33 - + dgam113*GamDDD322*invgam11*invgam33 - + dgam323*GamDDD122*invgam12*invgam33 - + dgam313*GamDDD222*invgam12*invgam33 - + dgam123*GamDDD322*invgam12*invgam33 - + dgam213*GamDDD322*invgam12*invgam33 - + dgam333*GamDDD122*invgam13*invgam33 - + dgam133*GamDDD322*invgam13*invgam33 - + 2*dgam313*GamDDD322*invgam13*invgam33 - + dgam323*GamDDD222*invgam22*invgam33 - + dgam223*GamDDD322*invgam22*invgam33 - + dgam333*GamDDD222*invgam23*invgam33 - + dgam233*GamDDD322*invgam23*invgam33 - + 2*dgam323*GamDDD322*invgam23*invgam33 - dgam333*GamDDD322*Power(invgam33,2) +; + +vreal tr1dGam23 += +(-(ddgam1123*invgam11) + ddgam1213*invgam11 + ddgam1312*invgam11 - + 2*dgam111*GamDDD123*Power(invgam11,2) - ddgam1223*invgam12 + + ddgam1322*invgam12 + ddgam2213*invgam12 + ddgam2312*invgam12 - + 4*dgam112*GamDDD123*invgam11*invgam12 - + 2*dgam211*GamDDD123*invgam11*invgam12 - + 2*dgam111*GamDDD223*invgam11*invgam12 - + 2*dgam122*GamDDD123*Power(invgam12,2) - + 2*dgam212*GamDDD123*Power(invgam12,2) - + 2*dgam112*GamDDD223*Power(invgam12,2) - + 2*dgam211*GamDDD223*Power(invgam12,2) + ddgam1233*invgam13 - + ddgam1323*invgam13 + ddgam2313*invgam13 + ddgam3312*invgam13 - + 4*dgam113*GamDDD123*invgam11*invgam13 - + 2*dgam311*GamDDD123*invgam11*invgam13 - + 2*dgam111*GamDDD323*invgam11*invgam13 - + 4*dgam123*GamDDD123*invgam12*invgam13 - + 2*dgam213*GamDDD123*invgam12*invgam13 - + 2*dgam312*GamDDD123*invgam12*invgam13 - + 2*dgam113*GamDDD223*invgam12*invgam13 - + 2*dgam311*GamDDD223*invgam12*invgam13 - + 2*dgam112*GamDDD323*invgam12*invgam13 - + 2*dgam211*GamDDD323*invgam12*invgam13 - + 2*dgam133*GamDDD123*Power(invgam13,2) - + 2*dgam313*GamDDD123*Power(invgam13,2) - + 2*dgam113*GamDDD323*Power(invgam13,2) - + 2*dgam311*GamDDD323*Power(invgam13,2) + ddgam2322*invgam22 - + 2*dgam212*GamDDD123*invgam11*invgam22 - + 2*dgam112*GamDDD223*invgam11*invgam22 - + 2*dgam222*GamDDD123*invgam12*invgam22 - + 2*dgam122*GamDDD223*invgam12*invgam22 - + 4*dgam212*GamDDD223*invgam12*invgam22 - + 2*dgam223*GamDDD123*invgam13*invgam22 - + 2*dgam123*GamDDD223*invgam13*invgam22 - + 2*dgam312*GamDDD223*invgam13*invgam22 - + 2*dgam212*GamDDD323*invgam13*invgam22 - + 2*dgam222*GamDDD223*Power(invgam22,2) + ddgam2233*invgam23 + + ddgam3322*invgam23 - 2*dgam213*GamDDD123*invgam11*invgam23 - + 2*dgam312*GamDDD123*invgam11*invgam23 - + 2*dgam113*GamDDD223*invgam11*invgam23 - + 2*dgam112*GamDDD323*invgam11*invgam23 - + 2*dgam223*GamDDD123*invgam12*invgam23 - + 2*dgam322*GamDDD123*invgam12*invgam23 - + 2*dgam123*GamDDD223*invgam12*invgam23 - + 4*dgam213*GamDDD223*invgam12*invgam23 - + 2*dgam312*GamDDD223*invgam12*invgam23 - + 2*dgam122*GamDDD323*invgam12*invgam23 - + 2*dgam212*GamDDD323*invgam12*invgam23 - + 2*dgam233*GamDDD123*invgam13*invgam23 - + 2*dgam323*GamDDD123*invgam13*invgam23 - + 2*dgam133*GamDDD223*invgam13*invgam23 - + 2*dgam313*GamDDD223*invgam13*invgam23 - + 2*dgam123*GamDDD323*invgam13*invgam23 - + 2*dgam213*GamDDD323*invgam13*invgam23 - + 4*dgam312*GamDDD323*invgam13*invgam23 - + 4*dgam223*GamDDD223*invgam22*invgam23 - + 2*dgam322*GamDDD223*invgam22*invgam23 - + 2*dgam222*GamDDD323*invgam22*invgam23 - + 2*dgam233*GamDDD223*Power(invgam23,2) - + 2*dgam323*GamDDD223*Power(invgam23,2) - + 2*dgam223*GamDDD323*Power(invgam23,2) - + 2*dgam322*GamDDD323*Power(invgam23,2) + ddgam2333*invgam33 - + 2*dgam313*GamDDD123*invgam11*invgam33 - + 2*dgam113*GamDDD323*invgam11*invgam33 - + 2*dgam323*GamDDD123*invgam12*invgam33 - + 2*dgam313*GamDDD223*invgam12*invgam33 - + 2*dgam123*GamDDD323*invgam12*invgam33 - + 2*dgam213*GamDDD323*invgam12*invgam33 - + 2*dgam333*GamDDD123*invgam13*invgam33 - + 2*dgam133*GamDDD323*invgam13*invgam33 - + 4*dgam313*GamDDD323*invgam13*invgam33 - + 2*dgam323*GamDDD223*invgam22*invgam33 - + 2*dgam223*GamDDD323*invgam22*invgam33 - + 2*dgam333*GamDDD223*invgam23*invgam33 - + 2*dgam233*GamDDD323*invgam23*invgam33 - + 4*dgam323*GamDDD323*invgam23*invgam33 - + 2*dgam333*GamDDD323*Power(invgam33,2))/2. +; + +vreal tr1dGam33 += +-0.5*(ddgam1133*invgam11) + ddgam1313*invgam11 - + dgam111*GamDDD133*Power(invgam11,2) - ddgam1233*invgam12 + + ddgam1323*invgam12 + ddgam2313*invgam12 - + 2*dgam112*GamDDD133*invgam11*invgam12 - + dgam211*GamDDD133*invgam11*invgam12 - + dgam111*GamDDD233*invgam11*invgam12 - + dgam122*GamDDD133*Power(invgam12,2) - + dgam212*GamDDD133*Power(invgam12,2) - + dgam112*GamDDD233*Power(invgam12,2) - + dgam211*GamDDD233*Power(invgam12,2) + ddgam3313*invgam13 - + 2*dgam113*GamDDD133*invgam11*invgam13 - + dgam311*GamDDD133*invgam11*invgam13 - + dgam111*GamDDD333*invgam11*invgam13 - + 2*dgam123*GamDDD133*invgam12*invgam13 - + dgam213*GamDDD133*invgam12*invgam13 - + dgam312*GamDDD133*invgam12*invgam13 - + dgam113*GamDDD233*invgam12*invgam13 - + dgam311*GamDDD233*invgam12*invgam13 - + dgam112*GamDDD333*invgam12*invgam13 - + dgam211*GamDDD333*invgam12*invgam13 - + dgam133*GamDDD133*Power(invgam13,2) - + dgam313*GamDDD133*Power(invgam13,2) - + dgam113*GamDDD333*Power(invgam13,2) - + dgam311*GamDDD333*Power(invgam13,2) - (ddgam2233*invgam22)/2. + + ddgam2323*invgam22 - dgam212*GamDDD133*invgam11*invgam22 - + dgam112*GamDDD233*invgam11*invgam22 - + dgam222*GamDDD133*invgam12*invgam22 - + dgam122*GamDDD233*invgam12*invgam22 - + 2*dgam212*GamDDD233*invgam12*invgam22 - + dgam223*GamDDD133*invgam13*invgam22 - + dgam123*GamDDD233*invgam13*invgam22 - + dgam312*GamDDD233*invgam13*invgam22 - + dgam212*GamDDD333*invgam13*invgam22 - + dgam222*GamDDD233*Power(invgam22,2) + ddgam3323*invgam23 - + dgam213*GamDDD133*invgam11*invgam23 - + dgam312*GamDDD133*invgam11*invgam23 - + dgam113*GamDDD233*invgam11*invgam23 - + dgam112*GamDDD333*invgam11*invgam23 - + dgam223*GamDDD133*invgam12*invgam23 - + dgam322*GamDDD133*invgam12*invgam23 - + dgam123*GamDDD233*invgam12*invgam23 - + 2*dgam213*GamDDD233*invgam12*invgam23 - + dgam312*GamDDD233*invgam12*invgam23 - + dgam122*GamDDD333*invgam12*invgam23 - + dgam212*GamDDD333*invgam12*invgam23 - + dgam233*GamDDD133*invgam13*invgam23 - + dgam323*GamDDD133*invgam13*invgam23 - + dgam133*GamDDD233*invgam13*invgam23 - + dgam313*GamDDD233*invgam13*invgam23 - + dgam123*GamDDD333*invgam13*invgam23 - + dgam213*GamDDD333*invgam13*invgam23 - + 2*dgam312*GamDDD333*invgam13*invgam23 - + 2*dgam223*GamDDD233*invgam22*invgam23 - + dgam322*GamDDD233*invgam22*invgam23 - + dgam222*GamDDD333*invgam22*invgam23 - + dgam233*GamDDD233*Power(invgam23,2) - + dgam323*GamDDD233*Power(invgam23,2) - + dgam223*GamDDD333*Power(invgam23,2) - + dgam322*GamDDD333*Power(invgam23,2) + (ddgam3333*invgam33)/2. - + dgam313*GamDDD133*invgam11*invgam33 - + dgam113*GamDDD333*invgam11*invgam33 - + dgam323*GamDDD133*invgam12*invgam33 - + dgam313*GamDDD233*invgam12*invgam33 - + dgam123*GamDDD333*invgam12*invgam33 - + dgam213*GamDDD333*invgam12*invgam33 - + dgam333*GamDDD133*invgam13*invgam33 - + dgam133*GamDDD333*invgam13*invgam33 - + 2*dgam313*GamDDD333*invgam13*invgam33 - + dgam323*GamDDD233*invgam22*invgam33 - + dgam223*GamDDD333*invgam22*invgam33 - + dgam333*GamDDD233*invgam23*invgam33 - + dgam233*GamDDD333*invgam23*invgam33 - + 2*dgam323*GamDDD333*invgam23*invgam33 - dgam333*GamDDD333*Power(invgam33,2) +; + +vreal tr2dGam11 += +(ddgam1111*invgam11 - Power(dgam111,2)*Power(invgam11,2) + + 2*ddgam1112*invgam12 - 2*Power(dgam112,2)*Power(invgam12,2) + + 2*ddgam1113*invgam13 - 4*dgam112*dgam113*invgam12*invgam13 - + 2*Power(dgam113,2)*Power(invgam13,2) - + 2*dgam111*(2*dgam112*invgam11*invgam12 + dgam122*Power(invgam12,2) + + invgam13*(2*dgam113*invgam11 + 2*dgam123*invgam12 + + dgam133*invgam13)) + ddgam1122*invgam22 - + 2*Power(dgam112,2)*invgam11*invgam22 - + 4*dgam112*dgam122*invgam12*invgam22 - + 4*dgam112*dgam123*invgam13*invgam22 - + Power(dgam122,2)*Power(invgam22,2) + 2*ddgam1123*invgam23 - + 4*dgam112*dgam113*invgam11*invgam23 - + 4*dgam113*dgam122*invgam12*invgam23 - + 4*dgam112*dgam123*invgam12*invgam23 - + 4*dgam113*dgam123*invgam13*invgam23 - + 4*dgam112*dgam133*invgam13*invgam23 - + 4*dgam122*dgam123*invgam22*invgam23 - + 2*Power(dgam123,2)*Power(invgam23,2) - + 2*dgam122*dgam133*Power(invgam23,2) + ddgam1133*invgam33 - + 2*Power(dgam113,2)*invgam11*invgam33 - + 4*dgam113*dgam123*invgam12*invgam33 - + 4*dgam113*dgam133*invgam13*invgam33 - + 2*Power(dgam123,2)*invgam22*invgam33 - + 4*dgam123*dgam133*invgam23*invgam33 - Power(dgam133,2)*Power(invgam33,2)\ +)/2. +; + +vreal tr2dGam12 += +(ddgam1211*invgam11 + 2*ddgam1212*invgam12 - + 2*dgam112*dgam211*invgam11*invgam12 - + dgam122*dgam211*Power(invgam12,2) - + 2*dgam112*dgam212*Power(invgam12,2) + 2*ddgam1213*invgam13 - + 2*dgam113*dgam211*invgam11*invgam13 - + 2*dgam123*dgam211*invgam12*invgam13 - + 2*dgam113*dgam212*invgam12*invgam13 - + 2*dgam112*dgam213*invgam12*invgam13 - + dgam133*dgam211*Power(invgam13,2) - + 2*dgam113*dgam213*Power(invgam13,2) - + dgam111*(dgam211*Power(invgam11,2) + 2*dgam212*invgam11*invgam12 + + dgam222*Power(invgam12,2) + 2*dgam213*invgam11*invgam13 + + 2*dgam223*invgam12*invgam13 + dgam233*Power(invgam13,2)) + + ddgam1222*invgam22 - 2*dgam112*dgam212*invgam11*invgam22 - + 2*dgam122*dgam212*invgam12*invgam22 - + 2*dgam112*dgam222*invgam12*invgam22 - + 2*dgam123*dgam212*invgam13*invgam22 - + 2*dgam112*dgam223*invgam13*invgam22 - + dgam122*dgam222*Power(invgam22,2) + 2*ddgam1223*invgam23 - + 2*dgam113*dgam212*invgam11*invgam23 - + 2*dgam112*dgam213*invgam11*invgam23 - + 2*dgam123*dgam212*invgam12*invgam23 - + 2*dgam122*dgam213*invgam12*invgam23 - + 2*dgam113*dgam222*invgam12*invgam23 - + 2*dgam112*dgam223*invgam12*invgam23 - + 2*dgam133*dgam212*invgam13*invgam23 - + 2*dgam123*dgam213*invgam13*invgam23 - + 2*dgam113*dgam223*invgam13*invgam23 - + 2*dgam112*dgam233*invgam13*invgam23 - + 2*dgam123*dgam222*invgam22*invgam23 - + 2*dgam122*dgam223*invgam22*invgam23 - + dgam133*dgam222*Power(invgam23,2) - + 2*dgam123*dgam223*Power(invgam23,2) - + dgam122*dgam233*Power(invgam23,2) + ddgam1233*invgam33 - + 2*dgam113*dgam213*invgam11*invgam33 - + 2*dgam123*dgam213*invgam12*invgam33 - + 2*dgam113*dgam223*invgam12*invgam33 - + 2*dgam133*dgam213*invgam13*invgam33 - + 2*dgam113*dgam233*invgam13*invgam33 - + 2*dgam123*dgam223*invgam22*invgam33 - + 2*dgam133*dgam223*invgam23*invgam33 - + 2*dgam123*dgam233*invgam23*invgam33 - dgam133*dgam233*Power(invgam33,2))/ + 2. +; + +vreal tr2dGam13 += +(ddgam1311*invgam11 + 2*ddgam1312*invgam12 - + 2*dgam112*dgam311*invgam11*invgam12 - + dgam122*dgam311*Power(invgam12,2) - + 2*dgam112*dgam312*Power(invgam12,2) + 2*ddgam1313*invgam13 - + 2*dgam113*dgam311*invgam11*invgam13 - + 2*dgam123*dgam311*invgam12*invgam13 - + 2*dgam113*dgam312*invgam12*invgam13 - + 2*dgam112*dgam313*invgam12*invgam13 - + dgam133*dgam311*Power(invgam13,2) - + 2*dgam113*dgam313*Power(invgam13,2) - + dgam111*(dgam311*Power(invgam11,2) + 2*dgam312*invgam11*invgam12 + + dgam322*Power(invgam12,2) + 2*dgam313*invgam11*invgam13 + + 2*dgam323*invgam12*invgam13 + dgam333*Power(invgam13,2)) + + ddgam1322*invgam22 - 2*dgam112*dgam312*invgam11*invgam22 - + 2*dgam122*dgam312*invgam12*invgam22 - + 2*dgam112*dgam322*invgam12*invgam22 - + 2*dgam123*dgam312*invgam13*invgam22 - + 2*dgam112*dgam323*invgam13*invgam22 - + dgam122*dgam322*Power(invgam22,2) + 2*ddgam1323*invgam23 - + 2*dgam113*dgam312*invgam11*invgam23 - + 2*dgam112*dgam313*invgam11*invgam23 - + 2*dgam123*dgam312*invgam12*invgam23 - + 2*dgam122*dgam313*invgam12*invgam23 - + 2*dgam113*dgam322*invgam12*invgam23 - + 2*dgam112*dgam323*invgam12*invgam23 - + 2*dgam133*dgam312*invgam13*invgam23 - + 2*dgam123*dgam313*invgam13*invgam23 - + 2*dgam113*dgam323*invgam13*invgam23 - + 2*dgam112*dgam333*invgam13*invgam23 - + 2*dgam123*dgam322*invgam22*invgam23 - + 2*dgam122*dgam323*invgam22*invgam23 - + dgam133*dgam322*Power(invgam23,2) - + 2*dgam123*dgam323*Power(invgam23,2) - + dgam122*dgam333*Power(invgam23,2) + ddgam1333*invgam33 - + 2*dgam113*dgam313*invgam11*invgam33 - + 2*dgam123*dgam313*invgam12*invgam33 - + 2*dgam113*dgam323*invgam12*invgam33 - + 2*dgam133*dgam313*invgam13*invgam33 - + 2*dgam113*dgam333*invgam13*invgam33 - + 2*dgam123*dgam323*invgam22*invgam33 - + 2*dgam133*dgam323*invgam23*invgam33 - + 2*dgam123*dgam333*invgam23*invgam33 - dgam133*dgam333*Power(invgam33,2))/ + 2. +; + +vreal tr2dGam22 += +(ddgam2211*invgam11 - Power(dgam211,2)*Power(invgam11,2) + + 2*ddgam2212*invgam12 - 2*Power(dgam212,2)*Power(invgam12,2) + + 2*ddgam2213*invgam13 - 4*dgam212*dgam213*invgam12*invgam13 - + 2*Power(dgam213,2)*Power(invgam13,2) - + 2*dgam211*(2*dgam212*invgam11*invgam12 + dgam222*Power(invgam12,2) + + invgam13*(2*dgam213*invgam11 + 2*dgam223*invgam12 + + dgam233*invgam13)) + ddgam2222*invgam22 - + 2*Power(dgam212,2)*invgam11*invgam22 - + 4*dgam212*dgam222*invgam12*invgam22 - + 4*dgam212*dgam223*invgam13*invgam22 - + Power(dgam222,2)*Power(invgam22,2) + 2*ddgam2223*invgam23 - + 4*dgam212*dgam213*invgam11*invgam23 - + 4*dgam213*dgam222*invgam12*invgam23 - + 4*dgam212*dgam223*invgam12*invgam23 - + 4*dgam213*dgam223*invgam13*invgam23 - + 4*dgam212*dgam233*invgam13*invgam23 - + 4*dgam222*dgam223*invgam22*invgam23 - + 2*Power(dgam223,2)*Power(invgam23,2) - + 2*dgam222*dgam233*Power(invgam23,2) + ddgam2233*invgam33 - + 2*Power(dgam213,2)*invgam11*invgam33 - + 4*dgam213*dgam223*invgam12*invgam33 - + 4*dgam213*dgam233*invgam13*invgam33 - + 2*Power(dgam223,2)*invgam22*invgam33 - + 4*dgam223*dgam233*invgam23*invgam33 - Power(dgam233,2)*Power(invgam33,2)\ +)/2. +; + +vreal tr2dGam23 += +(ddgam2311*invgam11 + 2*ddgam2312*invgam12 - + 2*dgam212*dgam311*invgam11*invgam12 - + dgam222*dgam311*Power(invgam12,2) - + 2*dgam212*dgam312*Power(invgam12,2) + 2*ddgam2313*invgam13 - + 2*dgam213*dgam311*invgam11*invgam13 - + 2*dgam223*dgam311*invgam12*invgam13 - + 2*dgam213*dgam312*invgam12*invgam13 - + 2*dgam212*dgam313*invgam12*invgam13 - + dgam233*dgam311*Power(invgam13,2) - + 2*dgam213*dgam313*Power(invgam13,2) - + dgam211*(dgam311*Power(invgam11,2) + 2*dgam312*invgam11*invgam12 + + dgam322*Power(invgam12,2) + 2*dgam313*invgam11*invgam13 + + 2*dgam323*invgam12*invgam13 + dgam333*Power(invgam13,2)) + + ddgam2322*invgam22 - 2*dgam212*dgam312*invgam11*invgam22 - + 2*dgam222*dgam312*invgam12*invgam22 - + 2*dgam212*dgam322*invgam12*invgam22 - + 2*dgam223*dgam312*invgam13*invgam22 - + 2*dgam212*dgam323*invgam13*invgam22 - + dgam222*dgam322*Power(invgam22,2) + 2*ddgam2323*invgam23 - + 2*dgam213*dgam312*invgam11*invgam23 - + 2*dgam212*dgam313*invgam11*invgam23 - + 2*dgam223*dgam312*invgam12*invgam23 - + 2*dgam222*dgam313*invgam12*invgam23 - + 2*dgam213*dgam322*invgam12*invgam23 - + 2*dgam212*dgam323*invgam12*invgam23 - + 2*dgam233*dgam312*invgam13*invgam23 - + 2*dgam223*dgam313*invgam13*invgam23 - + 2*dgam213*dgam323*invgam13*invgam23 - + 2*dgam212*dgam333*invgam13*invgam23 - + 2*dgam223*dgam322*invgam22*invgam23 - + 2*dgam222*dgam323*invgam22*invgam23 - + dgam233*dgam322*Power(invgam23,2) - + 2*dgam223*dgam323*Power(invgam23,2) - + dgam222*dgam333*Power(invgam23,2) + ddgam2333*invgam33 - + 2*dgam213*dgam313*invgam11*invgam33 - + 2*dgam223*dgam313*invgam12*invgam33 - + 2*dgam213*dgam323*invgam12*invgam33 - + 2*dgam233*dgam313*invgam13*invgam33 - + 2*dgam213*dgam333*invgam13*invgam33 - + 2*dgam223*dgam323*invgam22*invgam33 - + 2*dgam233*dgam323*invgam23*invgam33 - + 2*dgam223*dgam333*invgam23*invgam33 - dgam233*dgam333*Power(invgam33,2))/ + 2. +; + +vreal tr2dGam33 += +(ddgam3311*invgam11 - Power(dgam311,2)*Power(invgam11,2) + + 2*ddgam3312*invgam12 - 2*Power(dgam312,2)*Power(invgam12,2) + + 2*ddgam3313*invgam13 - 4*dgam312*dgam313*invgam12*invgam13 - + 2*Power(dgam313,2)*Power(invgam13,2) - + 2*dgam311*(2*dgam312*invgam11*invgam12 + dgam322*Power(invgam12,2) + + invgam13*(2*dgam313*invgam11 + 2*dgam323*invgam12 + + dgam333*invgam13)) + ddgam3322*invgam22 - + 2*Power(dgam312,2)*invgam11*invgam22 - + 4*dgam312*dgam322*invgam12*invgam22 - + 4*dgam312*dgam323*invgam13*invgam22 - + Power(dgam322,2)*Power(invgam22,2) + 2*ddgam3323*invgam23 - + 4*dgam312*dgam313*invgam11*invgam23 - + 4*dgam313*dgam322*invgam12*invgam23 - + 4*dgam312*dgam323*invgam12*invgam23 - + 4*dgam313*dgam323*invgam13*invgam23 - + 4*dgam312*dgam333*invgam13*invgam23 - + 4*dgam322*dgam323*invgam22*invgam23 - + 2*Power(dgam323,2)*Power(invgam23,2) - + 2*dgam322*dgam333*Power(invgam23,2) + ddgam3333*invgam33 - + 2*Power(dgam313,2)*invgam11*invgam33 - + 4*dgam313*dgam323*invgam12*invgam33 - + 4*dgam313*dgam333*invgam13*invgam33 - + 2*Power(dgam323,2)*invgam22*invgam33 - + 4*dgam323*dgam333*invgam23*invgam33 - Power(dgam333,2)*Power(invgam33,2)\ +)/2. +; + +vreal R11 += +-(Gam112*Gam211) - Power(Gam212,2) + Gam211*Gam222 - Gam113*Gam311 + + Gam223*Gam311 - 2*Gam213*Gam312 - Power(Gam313,2) + + Gam111*(Gam212 + Gam313) + Gam211*Gam323 + Gam311*Gam333 + tr1dGam11 - + tr2dGam11 +; + +vreal R12 += +-(Gam122*Gam211) - Gam123*Gam311 + Gam112*(Gam212 + Gam313) - + Gam213*Gam322 + Gam212*Gam323 - Gam313*Gam323 + Gam312*Gam333 + + tr1dGam12 - tr2dGam12 +; + +vreal R13 += +-(Gam123*Gam211) + Gam213*Gam222 - Gam212*Gam223 - Gam133*Gam311 - + Gam233*Gam312 + Gam223*Gam313 + Gam113*(Gam212 + Gam313) + tr1dGam13 - + tr2dGam13 +; + +vreal R22 += +-Power(Gam112,2) + Gam111*Gam122 - Gam122*Gam212 + Gam112*Gam222 - + 2*Gam123*Gam312 + Gam122*Gam313 + Gam113*Gam322 - Gam223*Gam322 + + Gam222*Gam323 - Power(Gam323,2) + Gam322*Gam333 + tr1dGam22 - tr2dGam22 +; + +vreal R23 += +Gam111*Gam123 - Gam122*Gam213 + Gam112*(-Gam113 + Gam223) - Gam133*Gam312 - + Gam233*Gam322 + Gam113*Gam323 + Gam223*Gam323 + tr1dGam23 - tr2dGam23 +; + +vreal R33 += +-Power(Gam113,2) + Gam111*Gam133 + Gam133*Gam212 - 2*Gam123*Gam213 - + Power(Gam223,2) + Gam112*Gam233 + Gam222*Gam233 - Gam133*Gam313 - + Gam233*Gam323 + Gam113*Gam333 + Gam223*Gam333 + tr1dGam33 - tr2dGam33 +; + +vreal trexK += +exK11*invgam11 + 2*exK12*invgam12 + 2*exK13*invgam13 + exK22*invgam22 + + 2*exK23*invgam23 + exK33*invgam33 +; + + +vreal DexK112 += +dexK112 - exK11*Gam112 - exK22*Gam211 - exK12*(Gam111 + Gam212) - + exK23*Gam311 - exK13*Gam312 +; + +vreal DexK113 += +dexK113 - exK11*Gam113 - exK23*Gam211 - exK12*Gam213 - exK33*Gam311 - + exK13*(Gam111 + Gam313) +; + +vreal DexK122 += +dexK122 - 2*(exK12*Gam112 + exK22*Gam212 + exK23*Gam312) +; + +vreal DexK123 += +dexK123 - exK13*Gam112 - exK12*Gam113 - exK23*Gam212 - exK22*Gam213 - + exK33*Gam312 - exK23*Gam313 +; + +vreal DexK133 += +dexK133 - 2*(exK13*Gam113 + exK23*Gam213 + exK33*Gam313) +; + +vreal DexK211 += +dexK211 - 2*(exK11*Gam112 + exK12*Gam212 + exK13*Gam312) +; + +vreal DexK212 += +dexK212 - exK11*Gam122 - exK22*Gam212 - exK12*(Gam112 + Gam222) - + exK23*Gam312 - exK13*Gam322 +; + +vreal DexK213 += +dexK213 - exK11*Gam123 - exK23*Gam212 - exK12*Gam223 - exK33*Gam312 - + exK13*(Gam112 + Gam323) +; + +vreal DexK223 += +dexK223 - exK13*Gam122 - exK12*Gam123 - exK23*Gam222 - exK22*Gam223 - + exK33*Gam322 - exK23*Gam323 +; + +vreal DexK233 += +dexK233 - 2*(exK13*Gam123 + exK23*Gam223 + exK33*Gam323) +; + +vreal DexK311 += +dexK311 - 2*(exK11*Gam113 + exK12*Gam213 + exK13*Gam313) +; + +vreal DexK312 += +dexK312 - exK11*Gam123 - exK22*Gam213 - exK12*(Gam113 + Gam223) - + exK23*Gam313 - exK13*Gam323 +; + +vreal DexK313 += +dexK313 - exK11*Gam133 - exK23*Gam213 - exK12*Gam233 - exK33*Gam313 - + exK13*(Gam113 + Gam333) +; + +vreal DexK322 += +dexK322 - 2*(exK12*Gam123 + exK22*Gam223 + exK23*Gam323) +; + +vreal DexK323 += +dexK323 - exK13*Gam123 - exK12*Gam133 - exK23*Gam223 - exK22*Gam233 - + exK33*Gam323 - exK23*Gam333 +; + + +vreal rho += +(Power(beta1,2)*eT11 + Power(beta2,2)*eT22 + 2*beta2*beta3*eT23 + + Power(beta3,2)*eT33 + 2*beta1*(beta2*eT12 + beta3*eT13 - eTt1) - + 2*beta2*eTt2 - 2*beta3*eTt3 + eTtt)/Power(alpha,2) +; + +vreal Sm1 += +(beta1*eT11 + beta2*eT12 + beta3*eT13 - eTt1)/alpha +; + +vreal Sm2 += +(beta1*eT12 + beta2*eT22 + beta3*eT23 - eTt2)/alpha +; + +vreal Sm3 += +(beta1*eT13 + beta2*eT23 + beta3*eT33 - eTt3)/alpha +; + + +local_HC.store(mask, index2, +-(Power(exK11,2)*Power(invgam11,2)) - 2*Power(exK13,2)*Power(invgam13,2) - + 2*exK11*(2*exK12*invgam11*invgam12 + exK22*Power(invgam12,2) + + invgam13*(2*exK13*invgam11 + 2*exK23*invgam12 + exK33*invgam13)) - + Power(exK22,2)*Power(invgam22,2) - + 2*Power(exK12,2)*(Power(invgam12,2) + invgam11*invgam22) - + 4*exK13*exK22*invgam12*invgam23 - 4*exK13*exK23*invgam13*invgam23 - + 4*exK22*exK23*invgam22*invgam23 - 2*Power(exK23,2)*Power(invgam23,2) - + 2*exK22*exK33*Power(invgam23,2) - + 4*exK12*(exK13*invgam12*invgam13 + exK22*invgam12*invgam22 + + exK23*invgam13*invgam22 + exK13*invgam11*invgam23 + + exK23*invgam12*invgam23 + exK33*invgam13*invgam23) - + 2*Power(exK13,2)*invgam11*invgam33 - 4*exK13*exK23*invgam12*invgam33 - + 4*exK13*exK33*invgam13*invgam33 - 2*Power(exK23,2)*invgam22*invgam33 - + 4*exK23*exK33*invgam23*invgam33 - Power(exK33,2)*Power(invgam33,2) + + invgam11*R11 + 2*invgam12*R12 + 2*invgam13*R13 + invgam22*R22 + + 2*invgam23*R23 + invgam33*R33 - 16*cpi*rho + Power(trexK,2) +); + +local_MC1.store(mask, index2, +-(DexK212*Power(invgam12,2)) + 2*DexK123*invgam12*invgam13 - + DexK213*invgam12*invgam13 - DexK312*invgam12*invgam13 + + DexK133*Power(invgam13,2) - DexK313*Power(invgam13,2) + + DexK212*invgam11*invgam22 + DexK223*invgam13*invgam22 - + DexK322*invgam13*invgam22 + DexK122* + (Power(invgam12,2) - invgam11*invgam22) - 2*DexK123*invgam11*invgam23 + + DexK213*invgam11*invgam23 + DexK312*invgam11*invgam23 - + DexK223*invgam12*invgam23 + DexK322*invgam12*invgam23 + + DexK233*invgam13*invgam23 - DexK323*invgam13*invgam23 - + DexK133*invgam11*invgam33 + DexK313*invgam11*invgam33 - + DexK233*invgam12*invgam33 + DexK323*invgam12*invgam33 - + 8*cpi*invgam11*Sm1 - 8*cpi*invgam12*Sm2 - 8*cpi*invgam13*Sm3 +); + +local_MC2.store(mask, index2, +-(DexK112*Power(invgam12,2)) + DexK211*Power(invgam12,2) - + DexK113*invgam12*invgam13 + DexK311*invgam12*invgam13 + + DexK112*invgam11*invgam22 - DexK211*invgam11*invgam22 + + DexK123*invgam13*invgam22 - 2*DexK213*invgam13*invgam22 + + DexK312*invgam13*invgam22 + DexK113*invgam11*invgam23 - + DexK311*invgam11*invgam23 - DexK123*invgam12*invgam23 + + 2*DexK213*invgam12*invgam23 - DexK312*invgam12*invgam23 + + DexK133*invgam13*invgam23 - DexK313*invgam13*invgam23 + + DexK233*Power(invgam23,2) - DexK323*Power(invgam23,2) - + DexK133*invgam12*invgam33 + DexK313*invgam12*invgam33 - + DexK233*invgam22*invgam33 + DexK323*invgam22*invgam33 - + 8*cpi*invgam12*Sm1 - 8*cpi*invgam22*Sm2 - 8*cpi*invgam23*Sm3 +); + +local_MC3.store(mask, index2, +-(DexK112*invgam12*invgam13) + DexK211*invgam12*invgam13 - + DexK113*Power(invgam13,2) + DexK311*Power(invgam13,2) - + DexK122*invgam13*invgam22 + DexK212*invgam13*invgam22 + + DexK112*invgam11*invgam23 - DexK211*invgam11*invgam23 + + DexK122*invgam12*invgam23 - DexK212*invgam12*invgam23 - + DexK123*invgam13*invgam23 - DexK213*invgam13*invgam23 + + 2*DexK312*invgam13*invgam23 - DexK223*Power(invgam23,2) + + DexK322*Power(invgam23,2) + DexK113*invgam11*invgam33 - + DexK311*invgam11*invgam33 + DexK123*invgam12*invgam33 + + DexK213*invgam12*invgam33 - 2*DexK312*invgam12*invgam33 + + DexK223*invgam22*invgam33 - DexK322*invgam22*invgam33 - + 8*cpi*invgam13*Sm1 - 8*cpi*invgam23*Sm2 - 8*cpi*invgam33*Sm3 +); + + + }); +}); + +#endif // #ifndef ADM_SET_CONSTRAINT_HXX + +/* ADM_set_constraint.hxx */ diff --git a/ADMconstraints/wolfram/ADM_set_constraint.wl b/ADMconstraints/wolfram/ADM_set_constraint.wl new file mode 100644 index 00000000..4fc5498c --- /dev/null +++ b/ADMconstraints/wolfram/ADM_set_constraint.wl @@ -0,0 +1,97 @@ +(* ::Package:: *) + +(* ADM_set_constraint.wl *) + +(* (c) Liwei Ji, 07/2024 *) + +Needs["xAct`xCoba`", FileNameJoin[{Environment["GENERATO"], "src/Generato.wl"}]] + +SetPVerbose[False]; + +SetPrintDate[False]; + +SetGridPointIndex[""]; + +SetTempVariableType["vreal"]; + +DefManifold[M3, 3, IndexRange[a, z]]; + +DefChart[cart, M3, {1, 2, 3}, {X[], Y[], Z[]}, ChartColor -> Blue]; + +(* Define Variables *) + +< detinvgam}; + SetEQNDelayed[detinvgam[], 1 / Det[Mat] // Simplify]; + SetEQNDelayed[invgam[i_, j_], invMat[[i[[1]], j[[1]]]] // Simplify] +]; + +SetOutputFile[FileNameJoin[{Directory[], "ADM_set_constraint.hxx"}]]; + +$MainPrint[] := + Module[{}, + PrintInitializations[{Mode -> "MainOut"}, ConstraintVarlist]; + pr[]; + + pr["noinline([&]() __attribute__((__flatten__, __hot__)) {"]; + pr[" grid.loop_int_device<0, 0, 0, vsize>("]; + pr[" grid.nghostzones, [=] ARITH_DEVICE(const PointDesc &p) ARITH_INLINE {"]; + pr[" const vbool mask = mask_for_loop_tail(p.i, p.imax);"]; + pr[" const GF3D2index index2(layout2, p.I);"]; + pr[" const GF3D5index index5(layout5, p.I);"]; + pr[]; + + PrintListInitializations[TmunuVarlist, "gf_", "index2"]; + PrintListInitializations[ADMVarlist, "tl_", "index5"]; + PrintListInitializations[dADMVarlist, "tl_", "index5"]; + PrintListInitializations[ddADMVarlist, "tl_", "index5"]; + pr[]; + + PrintInitializations[{Mode -> "MainIn"}, TmunuVarlist]; + PrintInitializations[{Mode -> "MainIn", StorageType -> "Tile"}, + ADMVarlist]; + PrintInitializations[{Mode -> "MainIn", StorageType -> "Tile", TensorType -> "Vect"}, + Drop[dADMVarlist, {-1}]]; + + (* treat dexK separately *) + Module[{printdK, varname, symmetry, printname}, + {varname, symmetry, printname} = ParseVar[dADMVarlist[[-1]]]; + printdK[kk_, ii_, jj_] := PrintComponentInitialization[{varname, symmetry}, dexK[{kk, -GetDefaultChart[]}, {ii, -GetDefaultChart[]}, {jj, -GetDefaultChart[]}]]; + SetParsePrintCompInitMode[MainIn -> True]; + SetParsePrintCompInitStorageType[Tile -> True]; + SetParsePrintCompInitTensorType[Vect -> True]; + Do[If[ii != jj || ii != kk || jj != kk, printdK[kk, ii, jj]], {kk, 1, 3}, {ii, 1, 3}, {jj, ii, 3}] + CleanParsePrintCompInitStorageType[]; + CleanParsePrintCompInitTensorType[]; + CleanParsePrintCompInitMode[]; + ]; + + PrintInitializations[{Mode -> "MainIn", StorageType -> "Tile", TensorType -> "Smat"}, + ddADMVarlist]; + pr[]; + PrintEquations[{Mode -> "Temp"}, Drop[IntermediateVarlist, {-1}]]; + pr[]; + + (* treat DexK separately *) + Module[{printDK}, + printDK[kk_, ii_, jj_] := PrintComponentEquation[GetDefaultChart[], DexK[{kk, -GetDefaultChart[]}, {ii, -GetDefaultChart[]}, {jj, -GetDefaultChart[]}]]; + SetParsePrintCompEQNMode[{NewVar -> True}]; + Do[If[ii != jj || ii != kk || jj != kk, printDK[kk, ii, jj]], {kk, 1, 3}, {ii, 1, 3}, {jj, ii, 3}] + CleanParsePrintCompEQNMode[]; + ]; + + pr[]; + PrintEquations[{Mode -> "Temp"}, MatterVarlist]; + pr[]; + PrintEquations[{Mode -> "Main"}, ConstraintVarlist]; + pr[]; + pr[" });"]; + pr["});"]; + ]; + +Import[FileNameJoin[{Environment["GENERATO"], "codes/CarpetX.wl"}]]; diff --git a/ADMconstraints/wolfram/check_rhs.ipynb b/ADMconstraints/wolfram/check_rhs.ipynb new file mode 100644 index 00000000..47319737 --- /dev/null +++ b/ADMconstraints/wolfram/check_rhs.ipynb @@ -0,0 +1,270 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "a265f5c5-fce9-45ad-b496-8bb1ee1f219b", + "metadata": {}, + "source": [ + "# ADM Equations" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "9642db04-5283-4582-99af-90ffff046096", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "------------------------------------------------------------\n", + "Package xAct`xPerm` version 1.2.3, {2015, 8, 23}\n", + "CopyRight (C) 2003-2020, Jose M. Martin-Garcia, under the General Public\\\n", + " \n", + "> License.\n", + "Connecting to external mac executable...\n", + "Connection established.\n", + "------------------------------------------------------------\n", + "Package xAct`xTensor` version 1.2.0, {2021, 10, 17}\n", + "CopyRight (C) 2002-2021, Jose M. Martin-Garcia, under the General Public\\\n", + " \n", + "> License.\n", + "------------------------------------------------------------\n", + "These packages come with ABSOLUTELY NO WARRANTY; for details type\\\n", + " \n", + "> Disclaimer[]. This is free software, and you are welcome to redistribute\\\n", + " \n", + "> it under certain conditions. See the General Public License for details.\n", + "------------------------------------------------------------\n", + "------------------------------------------------------------\n", + "Package xAct`xCoba` version 0.8.6, {2021, 2, 28}\n", + "CopyRight (C) 2005-2021, David Yllanes and Jose M. Martin-Garcia, under the\\\n", + " \n", + "> General Public License.\n", + "------------------------------------------------------------\n", + "These packages come with ABSOLUTELY NO WARRANTY; for details type\\\n", + " \n", + "> Disclaimer[]. This is free software, and you are welcome to redistribute\\\n", + " \n", + "> it under certain conditions. See the General Public License for details.\n", + "------------------------------------------------------------\n", + "------------------------------------------------------------\n", + "Package Generato`Basic`, {2024, 1, 11}\n", + "------------------------------------------------------------\n", + "------------------------------------------------------------\n", + "Package Generato`ParseMode`, {2024, 7, 06}\n", + "------------------------------------------------------------\n", + "------------------------------------------------------------\n", + "Package Generato`Component`, {2024, 1, 11}\n", + "------------------------------------------------------------\n", + "------------------------------------------------------------\n", + "Package Generato`Varlist`, {2024, 1, 11}\n", + "------------------------------------------------------------\n", + "------------------------------------------------------------\n", + "Package Generato`Interface`, {2024, 1, 11}\n", + "------------------------------------------------------------\n", + "------------------------------------------------------------\n", + "Package Generato`Derivation`, {2024, 1, 18}\n", + "------------------------------------------------------------\n", + "\n" + ] + } + ], + "source": [ + "Needs[\"xAct`xCoba`\", FileNameJoin[{Environment[\"GENERATO\"], \"src/Generato.wl\"}]]" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "ade74516-0ef1-4104-8217-d8968a1fa871", + "metadata": {}, + "outputs": [], + "source": [ + "DefManifold[M3, 3, IndexRange[a, z]];" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "d76d538a-d172-4043-a677-f39b0ba03467", + "metadata": {}, + "outputs": [], + "source": [ + "(* set mode to check equations *)\n", + "SetCheckInputEquations[True];" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "89c3c5eb-f13f-4be9-8117-c9e197d1114e", + "metadata": {}, + "outputs": [], + "source": [ + "<<\"wl/ADM_vars.wl\"" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "396420b3-c92c-49bf-acc6-b02599fffdd0", + "metadata": {}, + "outputs": [], + "source": [ + "<<\"wl/ADM_rhs.wl\"" + ] + }, + { + "cell_type": "markdown", + "id": "789279a3-122b-41b2-a9d7-e852493904d7", + "metadata": {}, + "source": [ + "## Check Expressions" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "07be83aa-230e-44b5-ba16-b6fc7df0626e", + "metadata": {}, + "outputs": [], + "source": [ + "rhsof[var_] := Module[{}, var /. {var[[0]] :> RHSOf[ToString[var[[0]]]]}];\n", + "eqssof[varlist_] := Module[{var}, \n", + " Table[var = varlist[[ivar]][[1]]; var == rhsof[var]//ScreenDollarIndices, {ivar, 1, Length[varlist]}]];" + ] + }, + { + "cell_type": "markdown", + "id": "fe765538-ea4c-48c4-8597-df4256c160c7", + "metadata": {}, + "source": [ + "### Intermediate Variables" + ] + }, + { + "cell_type": "markdown", + "id": "bbb20a49-7fcf-45bc-837c-77948b504af7", + "metadata": {}, + "source": [ + "#### Intermediate" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "de2b21c1-afc1-4772-af17-8b6b1d655f04", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
\"Output\"
" + ], + "text/plain": [ + "-Image-" + ] + }, + "execution_count": 9, + "metadata": { + "text/html": [], + "text/plain": [] + }, + "output_type": "execute_result" + } + ], + "source": [ + "Rasterize[Drop[eqssof[IntermediateVarlist], {1,2}]//MatrixForm]" + ] + }, + { + "cell_type": "markdown", + "id": "9ac1d847-e0fa-451c-9f6e-bac3670460e6", + "metadata": {}, + "source": [ + "#### Matter" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "5ddbd250-e7ec-437d-a69a-a1fc753dd032", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
\"Output\"
" + ], + "text/plain": [ + "-Image-" + ] + }, + "execution_count": 10, + "metadata": { + "text/html": [], + "text/plain": [] + }, + "output_type": "execute_result" + } + ], + "source": [ + "Rasterize[eqssof[MatterVarlist]//MatrixForm]" + ] + }, + { + "cell_type": "markdown", + "id": "b1c7f63e-2cb3-4340-a0b4-9403f8fe36b6", + "metadata": {}, + "source": [ + "#### Constraint" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "cbb63715-91d1-4e6d-ba35-7b40bf15078e", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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ADM_rhs.wl *) + +(* (c) Liwei Ji, 11/2024 *) + +(****************) + +(* Intermediate *) + +(****************) + +SetEQN[GamDDD[k_, i_, j_], 1/2 (dgam[i, j, k] + dgam[j, k, i] - dgam[k, i, j])]; + +SetEQN[Gam[k_, i_, j_], invgam[k, l] GamDDD[-l, i, j]]; + +SetEQN[tr1dGam[i_, j_], -invgam[k, p] invgam[l, q] dgam[-k, -p, -q] GamDDD[-l, i, j] + 1/2 invgam[k, l] (ddgam[-k, i, j, -l] + ddgam[-k, j, -l, i] - ddgam[-k, -l, i, j])]; + +SetEQN[tr2dGam[m_, i_], 1/2 (invgam[k, l] ddgam[m, i, -k, -l] - invgam[k, p] invgam[l, q] dgam[m, -p, -q] dgam[i, -k, -l])]; + +SetEQN[R[j_, k_], tr1dGam[j, k] - tr2dGam[j, k] + Gam[i, -i, -p] Gam[p, j, k] - Gam[i, j, -p] Gam[p, -i, k]]; + +SetEQN[trexK[], invgam[k, l] exK[-k, -l]]; + +SetEQN[DexK[k_, i_, j_], dexK[k, i, j] - Gam[l, k, i] exK[-l, j] - Gam[l, k, j] exK[-l, i]]; + +(* matter *) + +SetEQN[rho[], alpha[] ^ -2 (eTtt[] - 2 beta[j] eTt[-j] + beta[i] beta[j] eT[-i, -j])]; + +SetEQN[Sm[i_], -alpha[] ^ -1 (eTt[i] - beta[k] eT[-k, i])]; + +(***************) + +(* Constraints *) + +(***************) + +SetEQN[HC[], invgam[k, l] R[-k, -l] + trexK[] ^ 2 - invgam[i, k] invgam[j, l] exK[-i, -j] exK[-k, -l] - 16 cpi rho[]]; + +SetEQN[MC[i_], (invgam[i, k] invgam[j, l] - invgam[i, j] invgam[k, l]) DexK[-j, -k, -l] - 8 cpi invgam[i, j] Sm[-j]]; diff --git a/ADMconstraints/wolfram/wl/ADM_vars.wl b/ADMconstraints/wolfram/wl/ADM_vars.wl new file mode 100644 index 00000000..2f8e5a38 --- /dev/null +++ b/ADMconstraints/wolfram/wl/ADM_vars.wl @@ -0,0 +1,86 @@ +(* ::Package:: *) + +(* ADM_vars.wl *) + +(* (c) Liwei Ji, 07/2024 *) + +(* ADM variables *) + +(*******************) + +(* Input Variables *) + +(*******************) + +ADMVarlist = + GridTensors[ + {gam[-i, -j], Symmetric[{-i, -j}], PrintAs -> "\[Gamma]"}, + {exK[-i, -j], Symmetric[{-i, -j}], PrintAs -> "K"}, + {alpha[], PrintAs -> "\[Alpha]"}, + {beta[i], PrintAs -> "\[Beta]"} + ]; + +dADMVarlist = + GridTensors[ + {dgam[-k, -i, -j], Symmetric[{-i, -j}], PrintAs -> "\[PartialD]\[Gamma]"}, + {dexK[-k, -i, -j], Symmetric[{-i, -j}], PrintAs -> "\[PartialD]K"} + ]; + +ddADMVarlist = + GridTensors[ + {ddgam[-k, -l, -i, -j], GenSet[Cycles[{1, 2}], Cycles[{3, 4}]], PrintAs -> "\[PartialD]\[PartialD]\[Gamma]"} + ]; + +TmunuVarlist = + GridTensors[ + {eTtt[], PrintAs -> "\!\(\*SubscriptBox[\(T\), \(tt\)]\)"}, + {eTt[-i], PrintAs -> "\!\(\*SubscriptBox[\(T\), \(t\)]\)"}, + {eT[-i, -j], Symmetric[{-i, -j}], PrintAs -> "T"} + ]; + +(**************************) + +(* Intermediate Variables *) + +(**************************) + +IntermediateVarlist = + TempTensors[ + {detinvgam[], PrintAs -> "1/\[Gamma]"}, + {invgam[i, j], Symmetric[{i, j}], PrintAs -> "\[Gamma]"}, + {GamDDD [-k, -i, -j], Symmetric[{-i, -j}], PrintAs -> "\[CapitalGamma]"}, + {Gam [k, -i, -j], Symmetric[{-i, -j}], PrintAs -> "\[CapitalGamma]"}, + {tr1dGam[-i, -j], Symmetric[{-i, -j}], PrintAs -> "\[PartialD]\!\(\*SuperscriptBox[\(\[CapitalGamma]\), \((tr1)\)]\)"}, + {tr2dGam[-i, -j], Symmetric[{-i, -j}], PrintAs -> "\[PartialD]\!\(\*SuperscriptBox[\(\[CapitalGamma]\), \((tr2)\)]\)"}, + {R[-i, -j], Symmetric[{-i, -j}], PrintAs -> "R"}, + {trexK[], PrintAs -> "K"}, + {DexK[-k, -i, -j], Symmetric[{-i, -j}], PrintAs -> "DK"} + ]; + +(* Matter *) + +MatterVarlist = + TempTensors[ + {rho[], PrintAs -> "\[Rho]"}, + {Sm[-i], PrintAs -> "S"} + ]; + +(***************) + +(* Constraints *) + +(***************) + +ConstraintVarlist = + GridTensors[ + {HC[], PrintAs -> "H"}, + {MC[i], PrintAs -> "M"} + ]; + +(************) + +(* Constant *) + +(************) + +DefConstantSymbol[cpi, PrintAs -> "\[Pi]"];