ax_helm_kernel.h

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#ifndef __MATH_AX_HELM_KERNEL_H__
#define __MATH_AX_HELM_KERNEL_H__
/*
 Copyright (c) 2021-2024, The Neko Authors
 All rights reserved.
 
 Redistribution and use in source and binary forms, with or without
 modification, are permitted provided that the following conditions
 are met:
 
   * Redistributions of source code must retain the above copyright
     notice, this list of conditions and the following disclaimer.
 
   * Redistributions in binary form must reproduce the above
     copyright notice, this list of conditions and the following
     disclaimer in the documentation and/or other materials provided
     with the distribution.
 
   * Neither the name of the authors nor the names of its
     contributors may be used to endorse or promote products derived
     from this software without specific prior written permission.
 
 THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
 "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
 LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
 FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
 COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
 INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
 BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
 CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
 ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 POSSIBILITY OF SUCH DAMAGE.
*/
 
#include "elem_block.h"
 
/*
 * Elements per block for the vector kstep kernels, pinned rather than swept.
 *
 * These hold three times the register blocked state of the scalar ones -- six
 * T[LX] arrays rather than two -- and measure at 254-255 registers on sm_90
 * for every lx from 8 up, spilling at 10, 12, 14 and 16. There is no headroom
 * for a wider block, so they keep one element per block. Override with
 * -DNEKO_AX_HELM_VECTOR_EB_C=<0|1|2> if that ever changes.
 */
#ifndef NEKO_AX_HELM_VECTOR_EB_C
#define NEKO_AX_HELM_VECTOR_EB_C 0
#endif
 
template< typename T, const int LX, const int CHUNKS >
__global__ void ax_helm_kernel_1d(T * __restrict__ w,
                                  const T * __restrict__ u,
                                  const T * __restrict__ dx,
                                  const T * __restrict__ dy,
                                  const T * __restrict__ dz,
                                  const T * __restrict__ dxt,
                                  const T * __restrict__ dyt,
                                  const T * __restrict__ dzt,
                                  const T * __restrict__ h1,
                                  const T * __restrict__ g11,
                                  const T * __restrict__ g22,
                                  const T * __restrict__ g33,
                                  const T * __restrict__ g12,
                                  const T * __restrict__ g13,
                                  const T * __restrict__ g23) {
 
  __shared__ T shdx[LX*LX];
  __shared__ T shdy[LX*LX];
  __shared__ T shdzt[LX*LX];
 
  __shared__ T shdxt[LX*LX];
  __shared__ T shdyt[LX*LX];
  __shared__ T shdz[LX*LX];
 
  __shared__ T shu[LX*LX*LX];
  __shared__ T shur[LX*LX*LX];
  __shared__ T shus[LX*LX*LX];
  __shared__ T shut[LX*LX*LX];
 
  const int e = blockIdx.x;
  const int iii = threadIdx.x;
  const int nchunks = (LX * LX * LX - 1)/CHUNKS + 1;
 
  if (iii<LX*LX) {
    shdx[iii] = dx[iii];
    shdy[iii] = dy[iii];
    shdz[iii] = dz[iii];
  }
 
  {
    int i = iii;
    while (i < LX * LX * LX){
      shu[i] = u[i+e*LX*LX*LX];
      i = i + CHUNKS;
    }
  }
 
  __syncthreads();
 
  if (iii<LX*LX){
    shdxt[iii] = dxt[iii];
    shdyt[iii] = dyt[iii];
    shdzt[iii] = dzt[iii];
  }
 
  for (int n=0; n<nchunks; n++){
    const int ijk = iii+n*CHUNKS;
    const int jk = ijk/LX;
    const int i = ijk-jk*LX;
    const int k = jk/LX;
    const int j = jk-k*LX;
    if (i<LX && j<LX && k<LX && ijk < LX*LX*LX){
      const T G00 = g11[ijk+e*LX*LX*LX];
      const T G11 = g22[ijk+e*LX*LX*LX];
      const T G22 = g33[ijk+e*LX*LX*LX];
      const T G01 = g12[ijk+e*LX*LX*LX];
      const T G02 = g13[ijk+e*LX*LX*LX];
      const T G12 = g23[ijk+e*LX*LX*LX];
      const T H1 = h1[ijk+e*LX*LX*LX];
      T rtmp = 0.0;
      T stmp = 0.0;
      T ttmp = 0.0;
#pragma unroll
      for (int l = 0; l<LX; l++){
        rtmp = rtmp + shdx[i+l*LX] * shu[l+j*LX+k*LX*LX];
        stmp = stmp + shdy[j+l*LX] * shu[i+l*LX+k*LX*LX];
        ttmp = ttmp + shdz[k+l*LX] * shu[i+j*LX+l*LX*LX];
      }
      shur[ijk] = H1 * (G00 * rtmp + G01 * stmp + G02 * ttmp);
      shus[ijk] = H1 * (G01 * rtmp + G11 * stmp + G12 * ttmp);
      shut[ijk] = H1 * (G02 * rtmp + G12 * stmp + G22 * ttmp);
    }
  }
 
  __syncthreads();
 
  for (int n=0; n<nchunks; n++){
    const int ijk = iii+n*CHUNKS;
    const int jk = ijk/LX;
    const int k = jk/LX;
    const int j = jk-k*LX;
    const int i = ijk-jk*LX;
    if (i<LX && j<LX && k<LX && ijk <LX*LX*LX){
      T wijke = 0.0;
#pragma unroll
      for (int l = 0; l<LX; l++){
        wijke = wijke
              + shdxt[i+l*LX] * shur[l+j*LX+k*LX*LX]
              + shdyt[j+l*LX] * shus[i+l*LX+k*LX*LX]
              + shdzt[k+l*LX] * shut[i+j*LX+l*LX*LX];
      }
      w[ijk+e*LX*LX*LX] = wijke;
    }
  }
}
 
template< typename T, const int LX, const int EB >
__global__ void NEKO_EB_BOUNDS(LX*LX*EB)
ax_helm_kernel_kstep(T * __restrict__ w,
                     const T * __restrict__ u,
                     const T * __restrict__ dx,
                     const T * __restrict__ dy,
                     const T * __restrict__ dz,
                     const T * __restrict__ h1,
                     const T * __restrict__ g11,
                     const T * __restrict__ g22,
                     const T * __restrict__ g33,
                     const T * __restrict__ g12,
                     const T * __restrict__ g13,
                     const T * __restrict__ g23,
                     const int nelv) {
 
  /* Element independent, one copy per block */
  __shared__ T shdx[LX * LX];
  __shared__ T shdy[LX * LX];
  __shared__ T shdz[LX * LX];
 
  /* One slice per element in the block */
  __shared__ T shu[EB * LX * LX];
  __shared__ T shur[EB * LX * LX];
  __shared__ T shus[EB * LX * LX];
 
  static_assert(sizeof(shdx) +
                sizeof(shdy) +
                sizeof(shdz) +
                sizeof(shu) +
                sizeof(shur) +
                sizeof(shus)
                <= NEKO_EB_MAX_SMEM,
                "kstep block exceeds the shared memory budget");
 
  T ru[LX];
  T rw[LX];
  T rut;
 
  /* Threads past the last element still have to reach the block wide
     barriers below, so clamp their reads and drop their stores rather
     than returning early. At EB == 1 the grid covers nelv exactly, so all
     of this bookkeeping is constant folded and the kernel is exactly what
     it was before blocking */
  const int eb = (EB == 1) ? 0 : threadIdx.z;
  const int e_blk = blockIdx.x * EB + eb;
  const bool active = (EB == 1) ? true : (e_blk < nelv);
  const int e = active ? e_blk : (nelv - 1);
  const int j = threadIdx.y;
  const int i = threadIdx.x;
  const int ij = i + j*LX;
  const int sh = eb*LX*LX;
  const int ele = e*LX*LX*LX;
 
  if (eb == 0) {
    shdx[ij] = dx[ij];
    shdy[ij] = dy[ij];
    shdz[ij] = dz[ij];
  }
 
#pragma unroll
  for(int k = 0; k < LX; ++k){
    ru[k] = u[ij + k*LX*LX + ele];
    rw[k] = 0.0;
  }
 
 
  __syncthreads();
#pragma unroll
  for (int k = 0; k < LX; ++k){
    const int ijk = ij + k*LX*LX;
    const T G00 = g11[ijk+ele];
    const T G11 = g22[ijk+ele];
    const T G22 = g33[ijk+ele];
    const T G01 = g12[ijk+ele];
    const T G02 = g13[ijk+ele];
    const T G12 = g23[ijk+ele];
    const T H1  = h1[ijk+ele];
    T ttmp = 0.0;
    shu[sh + ij] = ru[k];
#pragma unroll
    for (int l = 0; l < LX; l++){
      ttmp += shdz[k+l*LX] * ru[l];
    }
    __syncthreads();
 
    T rtmp = 0.0;
    T stmp = 0.0;
#pragma unroll
    for (int l = 0; l < LX; l++){
      rtmp += shdx[i+l*LX] * shu[sh + l+j*LX];
      stmp += shdy[j+l*LX] * shu[sh + i+l*LX];
    }
    shur[sh + ij] = H1
                  * (G00 * rtmp
                     + G01 * stmp
                     + G02 * ttmp);
    shus[sh + ij] = H1
                  * (G01 * rtmp
                     + G11 * stmp
                     + G12 * ttmp);
    rut           = H1
                  * (G02 * rtmp
                     + G12 * stmp
                     + G22 * ttmp);
 
    __syncthreads();
 
    T wijke = 0.0;
#pragma unroll
    for (int l = 0; l < LX; l++){
      wijke += shur[sh + l+j*LX] * shdx[l+i*LX];
      rw[l] += rut * shdz[k+l*LX];
      wijke += shus[sh + i+l*LX] * shdy[l + j*LX];
    }
    rw[k] += wijke;
  }
  if (active) {
#pragma unroll
    for (int k = 0; k < LX; ++k){
      w[ij + k*LX*LX + ele] = rw[k];
    }
  }
}
 
template< typename T, const int LX, const int EB >
__global__ void NEKO_EB_BOUNDS(LX*LX*EB)
ax_helm_kernel_kstep_padded(T * __restrict__ w,
                            const T * __restrict__ u,
                            const T * __restrict__ dx,
                            const T * __restrict__ dy,
                            const T * __restrict__ dz,
                            const T * __restrict__ h1,
                            const T * __restrict__ g11,
                            const T * __restrict__ g22,
                            const T * __restrict__ g33,
                            const T * __restrict__ g12,
                            const T * __restrict__ g13,
                            const T * __restrict__ g23,
                            const int nelv) {
 
  /* Element independent, one copy per block */
  __shared__ T shdx[LX * (LX+1)];
  __shared__ T shdy[LX * (LX+1)];
  __shared__ T shdz[LX * (LX+1)];
 
  /* One slice per element in the block */
  __shared__ T shu[EB * LX * (LX+1)];
  __shared__ T shur[EB * LX * LX];  // only accessed using fastest dimension
  __shared__ T shus[EB * LX * (LX+1)];
 
  static_assert(sizeof(shdx) +
                sizeof(shdy) +
                sizeof(shdz) +
                sizeof(shu) +
                sizeof(shur) +
                sizeof(shus)
                <= NEKO_EB_MAX_SMEM,
                "kstep block exceeds the shared memory budget");
 
  T ru[LX];
  T rw[LX];
  T rut;
 
  /* At EB == 1 the grid covers nelv exactly, so the blocking bookkeeping is
     constant folded away and the kernel is exactly what it was before */
  const int eb = (EB == 1) ? 0 : threadIdx.z;
  const int e_blk = blockIdx.x * EB + eb;
  const bool active = (EB == 1) ? true : (e_blk < nelv);
  const int e = active ? e_blk : (nelv - 1);
  const int j = threadIdx.y;
  const int i = threadIdx.x;
  const int ij = i + j*LX;
  const int ij_p = i + j*(LX+1);
  const int sh = eb*LX*LX;
  const int sh_p = eb*LX*(LX+1);
  const int ele = e*LX*LX*LX;
 
  if (eb == 0) {
    shdx[ij_p] = dx[ij];
    shdy[ij_p] = dy[ij];
    shdz[ij_p] = dz[ij];
  }
 
#pragma unroll
  for(int k = 0; k < LX; ++k){
    ru[k] = u[ij + k*LX*LX + ele];
    rw[k] = 0.0;
  }
 
 
  __syncthreads();
#pragma unroll
  for (int k = 0; k < LX; ++k){
    const int ijk = ij + k*LX*LX;
    const T G00 = g11[ijk+ele];
    const T G11 = g22[ijk+ele];
    const T G22 = g33[ijk+ele];
    const T G01 = g12[ijk+ele];
    const T G02 = g13[ijk+ele];
    const T G12 = g23[ijk+ele];
    const T H1  = h1[ijk+ele];
    T ttmp = 0.0;
    shu[sh_p + ij_p] = ru[k];
#pragma unroll
    for (int l = 0; l < LX; l++){
      ttmp += shdz[k+l*(LX+1)] * ru[l];
    }
    __syncthreads();
 
    T rtmp = 0.0;
    T stmp = 0.0;
#pragma unroll
    for (int l = 0; l < LX; l++){
      rtmp += shdx[i+l*(LX+1)] * shu[sh_p + l+j*(LX+1)];
      stmp += shdy[j+l*(LX+1)] * shu[sh_p + i+l*(LX+1)];
    }
    shur[sh + ij]      = H1
                       * (G00 * rtmp
                          + G01 * stmp
                          + G02 * ttmp);
    shus[sh_p + ij_p]  = H1
                       * (G01 * rtmp
                          + G11 * stmp
                          + G12 * ttmp);
    rut                = H1
                       * (G02 * rtmp
                          + G12 * stmp
                          + G22 * ttmp);
 
    __syncthreads();
 
    T wijke = 0.0;
#pragma unroll
    for (int l = 0; l < LX; l++){
      wijke += shur[sh + l+j*LX] * shdx[l+i*(LX+1)];
      rw[l] += rut * shdz[k+l*(LX+1)];
      wijke += shus[sh_p + i+l*(LX+1)] * shdy[l + j*(LX+1)];
    }
    rw[k] += wijke;
  }
  if (active) {
#pragma unroll
    for (int k = 0; k < LX; ++k){
      w[ij + k*LX*LX + ele] = rw[k];
    }
  }
}
 
/*
 * Vector versions
 */
 
template< typename T, const int LX, const int EB >
__global__ void NEKO_EB_BOUNDS(LX*LX*EB)
ax_helm_kernel_vector_kstep(T * __restrict__ au,
                            T * __restrict__ av,
                            T * __restrict__ aw,
                            const T * __restrict__ u,
                            const T * __restrict__ v,
                            const T * __restrict__ w,
                            const T * __restrict__ dx,
                            const T * __restrict__ dy,
                            const T * __restrict__ dz,
                            const T * __restrict__ h1,
                            const T * __restrict__ g11,
                            const T * __restrict__ g22,
                            const T * __restrict__ g33,
                            const T * __restrict__ g12,
                            const T * __restrict__ g13,
                            const T * __restrict__ g23,
                            const int nelv) {
 
  /* Element independent, one copy per block */
  __shared__ T shdx[LX * LX];
  __shared__ T shdy[LX * LX];
  __shared__ T shdz[LX * LX];
 
  /* One slice per element in the block */
  __shared__ T shu[EB * LX * LX];
  __shared__ T shur[EB * LX * LX];
  __shared__ T shus[EB * LX * LX];
 
  __shared__ T shv[EB * LX * LX];
  __shared__ T shvr[EB * LX * LX];
  __shared__ T shvs[EB * LX * LX];
 
  __shared__ T shw[EB * LX * LX];
  __shared__ T shwr[EB * LX * LX];
  __shared__ T shws[EB * LX * LX];
 
  static_assert(sizeof(shdx) +
                sizeof(shdy) +
                sizeof(shdz) +
                sizeof(shu) +
                sizeof(shur) +
                sizeof(shus) +
                sizeof(shv) +
                sizeof(shvr) +
                sizeof(shvs) +
                sizeof(shw) +
                sizeof(shwr) +
                sizeof(shws)
                <= NEKO_EB_MAX_SMEM,
                "kstep block exceeds the shared memory budget");
 
  T ru[LX];
  T rv[LX];
  T rw[LX];
 
  T ruw[LX];
  T rvw[LX];
  T rww[LX];
 
  T rut;
  T rvt;
  T rwt;
 
  /* At EB == 1 the grid covers nelv exactly, so the blocking bookkeeping is
     constant folded away and the kernel is exactly what it was before */
  const int eb = (EB == 1) ? 0 : threadIdx.z;
  const int e_blk = blockIdx.x * EB + eb;
  const bool active = (EB == 1) ? true : (e_blk < nelv);
  const int e = active ? e_blk : (nelv - 1);
  const int j = threadIdx.y;
  const int i = threadIdx.x;
  const int ij = i + j*LX;
  const int sh = eb*LX*LX;
  const int ele = e*LX*LX*LX;
 
  if (eb == 0) {
    shdx[ij] = dx[ij];
    shdy[ij] = dy[ij];
    shdz[ij] = dz[ij];
  }
 
#pragma unroll
  for(int k = 0; k < LX; ++k){
    ru[k] = u[ij + k*LX*LX + ele];
    ruw[k] = 0.0;
 
    rv[k] = v[ij + k*LX*LX + ele];
    rvw[k] = 0.0;
 
    rw[k] = w[ij + k*LX*LX + ele];
    rww[k] = 0.0;
  }
 
 
  __syncthreads();
#pragma unroll
  for (int k = 0; k < LX; ++k){
    const int ijk = ij + k*LX*LX;
    const T G00 = g11[ijk+ele];
    const T G11 = g22[ijk+ele];
    const T G22 = g33[ijk+ele];
    const T G01 = g12[ijk+ele];
    const T G02 = g13[ijk+ele];
    const T G12 = g23[ijk+ele];
    const T H1  = h1[ijk+ele];
    T uttmp = 0.0;
    T vttmp = 0.0;
    T wttmp = 0.0;
    shu[sh + ij] = ru[k];
    shv[sh + ij] = rv[k];
    shw[sh + ij] = rw[k];
#pragma unroll
    for (int l = 0; l < LX; l++){
      uttmp += shdz[k+l*LX] * ru[l];
      vttmp += shdz[k+l*LX] * rv[l];
      wttmp += shdz[k+l*LX] * rw[l];
    }
    __syncthreads();
 
    T urtmp = 0.0;
    T ustmp = 0.0;
 
    T vrtmp = 0.0;
    T vstmp = 0.0;
 
    T wrtmp = 0.0;
    T wstmp = 0.0;
#pragma unroll
    for (int l = 0; l < LX; l++){
      urtmp += shdx[i+l*LX] * shu[sh + l+j*LX];
      ustmp += shdy[j+l*LX] * shu[sh + i+l*LX];
 
      vrtmp += shdx[i+l*LX] * shv[sh + l+j*LX];
      vstmp += shdy[j+l*LX] * shv[sh + i+l*LX];
 
      wrtmp += shdx[i+l*LX] * shw[sh + l+j*LX];
      wstmp += shdy[j+l*LX] * shw[sh + i+l*LX];
    }
 
    shur[sh + ij] = H1
                  * (G00 * urtmp
                     + G01 * ustmp
                     + G02 * uttmp);
    shus[sh + ij] = H1
                  * (G01 * urtmp
                     + G11 * ustmp
                     + G12 * uttmp);
    rut           = H1
                  * (G02 * urtmp
                     + G12 * ustmp
                     + G22 * uttmp);
 
    shvr[sh + ij] = H1
                  * (G00 * vrtmp
                     + G01 * vstmp
                     + G02 * vttmp);
    shvs[sh + ij] = H1
                  * (G01 * vrtmp
                     + G11 * vstmp
                     + G12 * vttmp);
    rvt           = H1
                  * (G02 * vrtmp
                     + G12 * vstmp
                     + G22 * vttmp);
 
    shwr[sh + ij] = H1
                  * (G00 * wrtmp
                     + G01 * wstmp
                     + G02 * wttmp);
    shws[sh + ij] = H1
                  * (G01 * wrtmp
                     + G11 * wstmp
                     + G12 * wttmp);
    rwt           = H1
                  * (G02 * wrtmp
                     + G12 * wstmp
                     + G22 * wttmp);
 
    __syncthreads();
 
    T uwijke = 0.0;
    T vwijke = 0.0;
    T wwijke = 0.0;
#pragma unroll
    for (int l = 0; l < LX; l++){
      uwijke += shur[sh + l+j*LX] * shdx[l+i*LX];
      ruw[l] += rut * shdz[k+l*LX];
      uwijke += shus[sh + i+l*LX] * shdy[l + j*LX];
 
      vwijke += shvr[sh + l+j*LX] * shdx[l+i*LX];
      rvw[l] += rvt * shdz[k+l*LX];
      vwijke += shvs[sh + i+l*LX] * shdy[l + j*LX];
 
      wwijke += shwr[sh + l+j*LX] * shdx[l+i*LX];
      rww[l] += rwt * shdz[k+l*LX];
      wwijke += shws[sh + i+l*LX] * shdy[l + j*LX];
    }
    ruw[k] += uwijke;
    rvw[k] += vwijke;
    rww[k] += wwijke;
  }
  if (active) {
#pragma unroll
    for (int k = 0; k < LX; ++k){
      au[ij + k*LX*LX + ele] = ruw[k];
      av[ij + k*LX*LX + ele] = rvw[k];
      aw[ij + k*LX*LX + ele] = rww[k];
    }
  }
}
 
template< typename T, const int LX, const int EB >
__global__ void NEKO_EB_BOUNDS(LX*LX*EB)
ax_helm_kernel_vector_kstep_padded(T * __restrict__ au,
                                   T * __restrict__ av,
                                   T * __restrict__ aw,
                                   const T * __restrict__ u,
                                   const T * __restrict__ v,
                                   const T * __restrict__ w,
                                   const T * __restrict__ dx,
                                   const T * __restrict__ dy,
                                   const T * __restrict__ dz,
                                   const T * __restrict__ h1,
                                   const T * __restrict__ g11,
                                   const T * __restrict__ g22,
                                   const T * __restrict__ g33,
                                   const T * __restrict__ g12,
                                   const T * __restrict__ g13,
                                   const T * __restrict__ g23,
                                   const int nelv) {
 
  /* Element independent, one copy per block */
  __shared__ T shdx[LX * (LX+1)];
  __shared__ T shdy[LX * (LX+1)];
  __shared__ T shdz[LX * (LX+1)];
 
  /* One slice per element in the block */
  __shared__ T shu[EB * LX * (LX+1)];
  __shared__ T shur[EB * LX * LX];
  __shared__ T shus[EB * LX * (LX+1)];
 
  __shared__ T shv[EB * LX * (LX+1)];
  __shared__ T shvr[EB * LX * LX];
  __shared__ T shvs[EB * LX * (LX+1)];
 
  __shared__ T shw[EB * LX * (LX+1)];
  __shared__ T shwr[EB * LX * LX];
  __shared__ T shws[EB * LX * (LX+1)];
 
  static_assert(sizeof(shdx) +
                sizeof(shdy) +
                sizeof(shdz) +
                sizeof(shu) +
                sizeof(shur) +
                sizeof(shus) +
                sizeof(shv) +
                sizeof(shvr) +
                sizeof(shvs) +
                sizeof(shw) +
                sizeof(shwr) +
                sizeof(shws)
                <= NEKO_EB_MAX_SMEM,
                "kstep block exceeds the shared memory budget");
 
  T ru[LX];
  T rv[LX];
  T rw[LX];
 
  T ruw[LX];
  T rvw[LX];
  T rww[LX];
 
  T rut;
  T rvt;
  T rwt;
 
  /* At EB == 1 the grid covers nelv exactly, so the blocking bookkeeping is
     constant folded away and the kernel is exactly what it was before */
  const int eb = (EB == 1) ? 0 : threadIdx.z;
  const int e_blk = blockIdx.x * EB + eb;
  const bool active = (EB == 1) ? true : (e_blk < nelv);
  const int e = active ? e_blk : (nelv - 1);
  const int j = threadIdx.y;
  const int i = threadIdx.x;
  const int ij = i + j*LX;
  const int ij_p = i + j*(LX+1);
  const int sh = eb*LX*LX;
  const int sh_p = eb*LX*(LX+1);
  const int ele = e*LX*LX*LX;
 
  if (eb == 0) {
    shdx[ij_p] = dx[ij];
    shdy[ij_p] = dy[ij];
    shdz[ij_p] = dz[ij];
  }
 
#pragma unroll
  for(int k = 0; k < LX; ++k){
    ru[k] = u[ij + k*LX*LX + ele];
    ruw[k] = 0.0;
 
    rv[k] = v[ij + k*LX*LX + ele];
    rvw[k] = 0.0;
 
    rw[k] = w[ij + k*LX*LX + ele];
    rww[k] = 0.0;
  }
 
 
  __syncthreads();
#pragma unroll
  for (int k = 0; k < LX; ++k){
    const int ijk = ij + k*LX*LX;
    const T G00 = g11[ijk+ele];
    const T G11 = g22[ijk+ele];
    const T G22 = g33[ijk+ele];
    const T G01 = g12[ijk+ele];
    const T G02 = g13[ijk+ele];
    const T G12 = g23[ijk+ele];
    const T H1  = h1[ijk+ele];
    T uttmp = 0.0;
    T vttmp = 0.0;
    T wttmp = 0.0;
    shu[sh_p + ij_p] = ru[k];
    shv[sh_p + ij_p] = rv[k];
    shw[sh_p + ij_p] = rw[k];
#pragma unroll
    for (int l = 0; l < LX; l++){
      uttmp += shdz[k+l*(LX+1)] * ru[l];
      vttmp += shdz[k+l*(LX+1)] * rv[l];
      wttmp += shdz[k+l*(LX+1)] * rw[l];
    }
    __syncthreads();
 
    T urtmp = 0.0;
    T ustmp = 0.0;
 
    T vrtmp = 0.0;
    T vstmp = 0.0;
 
    T wrtmp = 0.0;
    T wstmp = 0.0;
#pragma unroll
    for (int l = 0; l < LX; l++){
      urtmp += shdx[i+l*(LX+1)] * shu[sh_p + l+j*(LX+1)];
      ustmp += shdy[j+l*(LX+1)] * shu[sh_p + i+l*(LX+1)];
 
      vrtmp += shdx[i+l*(LX+1)] * shv[sh_p + l+j*(LX+1)];
      vstmp += shdy[j+l*(LX+1)] * shv[sh_p + i+l*(LX+1)];
 
      wrtmp += shdx[i+l*(LX+1)] * shw[sh_p + l+j*(LX+1)];
      wstmp += shdy[j+l*(LX+1)] * shw[sh_p + i+l*(LX+1)];
    }
 
    shur[sh + ij]     = H1
                      * (G00 * urtmp
                         + G01 * ustmp
                         + G02 * uttmp);
    shus[sh_p + ij_p] = H1
                      * (G01 * urtmp
                         + G11 * ustmp
                         + G12 * uttmp);
    rut               = H1
                      * (G02 * urtmp
                         + G12 * ustmp
                         + G22 * uttmp);
 
    shvr[sh + ij]     = H1
                      * (G00 * vrtmp
                         + G01 * vstmp
                         + G02 * vttmp);
    shvs[sh_p + ij_p] = H1
                      * (G01 * vrtmp
                         + G11 * vstmp
                         + G12 * vttmp);
    rvt               = H1
                      * (G02 * vrtmp
                         + G12 * vstmp
                         + G22 * vttmp);
 
    shwr[sh + ij]     = H1
                      * (G00 * wrtmp
                         + G01 * wstmp
                         + G02 * wttmp);
    shws[sh_p + ij_p] = H1
                      * (G01 * wrtmp
                         + G11 * wstmp
                         + G12 * wttmp);
    rwt               = H1
                      * (G02 * wrtmp
                         + G12 * wstmp
                         + G22 * wttmp);
 
    __syncthreads();
 
    T uwijke = 0.0;
    T vwijke = 0.0;
    T wwijke = 0.0;
#pragma unroll
    for (int l = 0; l < LX; l++){
      uwijke += shur[sh + l+j*LX] * shdx[l+i*(LX+1)];
      ruw[l] += rut * shdz[k+l*(LX+1)];
      uwijke += shus[sh_p + i+l*(LX+1)] * shdy[l + j*(LX+1)];
 
      vwijke += shvr[sh + l+j*LX] * shdx[l+i*(LX+1)];
      rvw[l] += rvt * shdz[k+l*(LX+1)];
      vwijke += shvs[sh_p + i+l*(LX+1)] * shdy[l + j*(LX+1)];
 
      wwijke += shwr[sh + l+j*LX] * shdx[l+i*(LX+1)];
      rww[l] += rwt * shdz[k+l*(LX+1)];
      wwijke += shws[sh_p + i+l*(LX+1)] * shdy[l + j*(LX+1)];
    }
    ruw[k] += uwijke;
    rvw[k] += vwijke;
    rww[k] += wwijke;
  }
  if (active) {
#pragma unroll
    for (int k = 0; k < LX; ++k){
      au[ij + k*LX*LX + ele] = ruw[k];
      av[ij + k*LX*LX + ele] = rvw[k];
      aw[ij + k*LX*LX + ele] = rww[k];
    }
  }
}
 
template< typename T >
__global__ void ax_helm_kernel_vector_part2(T * __restrict__ au,
                                            T * __restrict__ av,
                                            T * __restrict__ aw,
                                            const T * __restrict__ u,
                                            const T * __restrict__ v,
                                            const T * __restrict__ w,
                                            const T * __restrict__ h2,
                                            const T * __restrict__ B,
                                            const int n) {
 
  const int idx = blockIdx.x * blockDim.x + threadIdx.x;
  const int str = blockDim.x * gridDim.x;
 
  for (int i = idx; i < n; i += str) {
    au[i] = au[i] + h2[i] * B[i] * u[i];
    av[i] = av[i] + h2[i] * B[i] * v[i];
    aw[i] = aw[i] + h2[i] * B[i] * w[i];
  }
 
}
#endif // __MATH_AX_HELM_KERNEL_H__