FFT: OpenBC Solver

This implements the algorithm of Hockney, Methods Comp. Phys. 9 (1970),
136-210 for solving Poisson's equation with open boundaries.

Author: WeiqunZhang

Docs/sphinx_documentation/source/FFT.rst

@@ -71,9 +71,14 @@ object. Therefore, one should cache it for reuse if possible. Although
 Poisson Solver
 ==============
 
-AMReX provides FFT based Poisson solvers. :cpp:`FFT::Poisson` supports
-periodic (:cpp:`FFT::Boundary::periodic`), homogeneous Neumann
-(:cpp:`FFT::Boundary::even`), and homogeneous Dirichlet
+AMReX provides FFT based Poisson solvers. Here, Poisson's equation is
+
+.. math::
+
+  \nabla^2 \phi = \rho.
+
+:cpp:`FFT::Poisson` supports periodic (:cpp:`FFT::Boundary::periodic`),
+homogeneous Neumann (:cpp:`FFT::Boundary::even`), and homogeneous Dirichlet
 (:cpp:`FFT::Boundary::odd`) boundaries using FFT. Below is an example of
 using the solver.
 
@@ -103,6 +108,27 @@ using the solver.
     FFT::Poisson fft_poisson(geom, fft_bc);
     fft_poisson.solve(soln, rhs);
 
+:cpp:`FFT::PoissonOpenBC` is a 3D only solver that supports open
+boundaries. Its implementation utilizes :cpp:`FFT::OpenBCSolver`, which can
+be used for implementing convolution based solvers with a user provided
+Green's function. If users want to extend the open BC solver to 2D or other
+types of Green's function, they could use :cpp:`FFT::PoissonOpenBC` as an
+example. Below is an example of solving Poisson's equation with open
+boundaries.
+
+.. highlight:: c++
+
+::
+
+    Geometry geom(...);
+    MultiFab soln(...); // soln can be either nodal or cell-centered.
+    MultiFab rhs(...);  // rhs must have the same index type as soln.
+
+    int ng = ...; // ng can be non-zero, if we want to compute potential
+                  // outside the domain.
+    FFT::PoissonOpenBC openbc_solver(geom, soln.ixType(), IntVect(ng));
+    openbc_solver.solve(soln, rhs);
+
 :cpp:`FFT::PoissonHybrid` is a 3D only solver that supports periodic
 boundaries in the first two dimensions and Neumann boundary in the last
 dimension. The last dimension is solved with a tridiagonal solver that can

Src/Base/AMReX_BoxArray.H

@@ -69,7 +69,8 @@ namespace amrex
      */
     [[nodiscard]] BoxArray decompose (Box const& domain, int nboxes,
                                       Array<bool,AMREX_SPACEDIM> const& decomp
-                                      = {AMREX_D_DECL(true,true,true)});
+                                      = {AMREX_D_DECL(true,true,true)},
+                                      bool no_overlap = false);
 
 struct BARef
 {

Src/Base/AMReX_BoxArray.cpp

@@ -1891,7 +1891,7 @@ bool match (const BoxArray& x, const BoxArray& y)
 }
 
 BoxArray decompose (Box const& domain, int nboxes,
-                    Array<bool,AMREX_SPACEDIM> const& decomp)
+                    Array<bool,AMREX_SPACEDIM> const& decomp, bool no_overlap)
 {
     auto ndecomp = std::count(decomp.begin(), decomp.end(), true);
 
@@ -2048,9 +2048,24 @@ BoxArray decompose (Box const& domain, int nboxes,
                 ilo += domlo[0];
                 ihi += domlo[0];
                 Box b{IntVect(AMREX_D_DECL(ilo,jlo,klo)),
-                      IntVect(AMREX_D_DECL(ihi,jhi,khi))};
+                      IntVect(AMREX_D_DECL(ihi,jhi,khi)), ixtyp};
                 if (b.ok()) {
-                    bl.push_back(b.convert(ixtyp));
+                    if (no_overlap) {
+                        for (int idim = 0; idim < AMREX_SPACEDIM; ++idim) {
+                            if (ixtyp.nodeCentered(idim) &&
+                                b.bigEnd(idim) == ccdomain.bigEnd(idim))
+                            {
+                                b.growHi(idim, 1);
+                            }
+                        }
+                    } else {
+                        for (int idim = 0; idim < AMREX_SPACEDIM; ++idim) {
+                            if (ixtyp.nodeCentered(idim)) {
+                                b.growHi(idim, 1);
+                            }
+                        }
+                    }
+                    bl.push_back(b);
                 }
     AMREX_D_TERM(},},})
 

Src/FFT/AMReX_FFT.H

@@ -2,6 +2,7 @@
 #define AMREX_FFT_H_
 #include <AMReX_Config.H>
 
+#include <AMReX_FFT_OpenBCSolver.H>
 #include <AMReX_FFT_R2C.H>
 #include <AMReX_FFT_R2X.H>
 

Src/FFT/AMReX_FFT_OpenBCSolver.H

@@ -0,0 +1,142 @@
+#ifndef AMREX_FFT_OPENBC_SOLVER_H_
+#define AMREX_FFT_OPENBC_SOLVER_H_
+
+#include <AMReX_FFT_R2C.H>
+
+#include <AMReX_VisMF.H>
+
+namespace amrex::FFT
+{
+
+template <typename T = Real>
+class OpenBCSolver
+{
+public:
+    using MF = typename R2C<T>::MF;
+    using cMF = typename R2C<T>::cMF;
+
+    explicit OpenBCSolver (Box const& domain);
+
+    template <class F>
+    void setGreensFunction (F const& greens_function);
+
+    void solve (MF& phi, MF const& rho);
+
+    [[nodiscard]] Box const& Domain () const { return m_domain; }
+
+private:
+    Box m_domain;
+    R2C<T> m_r2c;
+    cMF m_G_fft;
+};
+
+template <typename T>
+OpenBCSolver<T>::OpenBCSolver (Box const& domain)
+    : m_domain(domain),
+      m_r2c(Box(domain.smallEnd(), domain.bigEnd()+domain.length(), domain.ixType()))
+{
+    auto [sd, ord] = m_r2c.getSpectralData();
+    amrex::ignore_unused(ord);
+    m_G_fft.define(sd->boxArray(), sd->DistributionMap(), 1, 0);
+}
+
+template <typename T>
+template <class F>
+void OpenBCSolver<T>::setGreensFunction (F const& greens_function)
+{
+    auto* infab = detail::get_fab(m_r2c.m_rx);
+    auto const& lo = m_domain.smallEnd();
+    auto const& lo3 = lo.dim3();
+    auto const& len = m_domain.length3d();
+    if (infab) {
+        auto const& a = infab->array();
+        auto box = infab->box();
+        GpuArray<int,3> nimages{1,1,1};
+        for (int idim = 0; idim < AMREX_SPACEDIM; ++idim) {
+            if (box.smallEnd(idim) == lo[idim] && box.length(idim) == 2*len[idim]) {
+                box.growHi(idim, -len[idim]+1); // +1 to include the middle plane
+                nimages[idim] = 2;
+            }
+        }
+        AMREX_ASSERT(nimages[0] == 2);
+        box.shift(-lo);
+        amrex::ParallelFor(box, [=] AMREX_GPU_DEVICE (int i, int j, int k)
+        {
+            if (i == len[0] || j == len[1] || k == len[2]) {
+                a(i+lo3.x,j+lo3.y,k+lo3.z) = T(0);
+            } else {
+                auto ii = i;
+                auto jj = (j > len[1]) ? 2*len[1]-j : j;
+                auto kk = (k > len[2]) ? 2*len[2]-k : k;
+                auto G = greens_function(ii+lo3.x,jj+lo3.y,kk+lo3.z);
+                for (int koff = 0; koff < nimages[2]; ++koff) {
+                    int k2 = (koff == 0) ?  k : 2*len[2]-k;
+                    if (k2 == 2*len[2]) { continue; }
+                    for (int joff = 0; joff < nimages[1]; ++joff) {
+                        int j2 = (joff == 0) ?  j : 2*len[1]-j;
+                        if (j2 == 2*len[1]) { continue; }
+                        for (int ioff = 0; ioff < nimages[0]; ++ioff) {
+                            int i2 = (ioff == 0) ?  i : 2*len[0]-i;
+                            if (i2 == 2*len[0]) { continue; }
+                            a(i2+lo3.x,j2+lo3.y,k2+lo3.z) = G;
+                        }
+                    }
+                }
+            }
+        });
+    }
+
+    m_r2c.forward(m_r2c.m_rx);
+
+    auto [sd, ord] = m_r2c.getSpectralData();
+    amrex::ignore_unused(ord);
+    auto const* srcfab = detail::get_fab(*sd);
+    if (srcfab) {
+        auto* dstfab = detail::get_fab(m_G_fft);
+        if (dstfab) {
+#if defined(AMREX_USE_GPU)
+            Gpu::dtod_memcpy_async
+#else
+            std::memcpy
+#endif
+                (dstfab->dataPtr(), srcfab->dataPtr(), dstfab->nBytes());
+        } else {
+            amrex::Abort("FFT::OpenBCSolver: how did this happen");
+        }
+    }
+}
+
+template <typename T>
+void OpenBCSolver<T>::solve (MF& phi, MF const& rho)
+{
+    auto& inmf = m_r2c.m_rx;
+    inmf.setVal(T(0));
+    inmf.ParallelCopy(rho, 0, 0, 1);
+
+    m_r2c.forward(inmf);
+
+    auto scaling_factor = T(1) / T(m_r2c.m_real_domain.numPts());
+
+    auto const* gfab = detail::get_fab(m_G_fft);
+    if (gfab) {
+        auto [sd, ord] = m_r2c.getSpectralData();
+        amrex::ignore_unused(ord);
+        auto* rhofab = detail::get_fab(*sd);
+        if (rhofab) {
+            auto* pdst = rhofab->dataPtr();
+            auto const* psrc = gfab->dataPtr();
+            amrex::ParallelFor(rhofab->box().numPts(), [=] AMREX_GPU_DEVICE (Long i)
+            {
+                pdst[i] *= psrc[i] * scaling_factor;
+            });
+        } else {
+            amrex::Abort("FFT::OpenBCSolver::solve: how did this happen?");
+        }
+    }
+
+    m_r2c.backward_doit(phi, phi.nGrowVect());
+}
+
+}
+
+#endif

Src/FFT/AMReX_FFT_Poisson.H

@@ -8,7 +8,8 @@ namespace amrex::FFT
 {
 
 /**
- * \brief Poisson solver for all periodic boundaries using FFT
+ * \brief Poisson solver for periodic, Dirichlet & Neumann boundaries using
+ * FFT.
  */
 template <typename MF = MultiFab>
 class Poisson
@@ -40,6 +41,32 @@ private:
     R2X<typename MF::value_type> m_r2x;
 };
 
+#if (AMREX_SPACEDIM == 3)
+/**
+ * \brief Poisson solve for Open BC using FFT.
+ */
+template <typename MF = MultiFab>
+class PoissonOpenBC
+{
+public:
+
+    template <typename FA=MF, std::enable_if_t<IsFabArray_v<FA>,int> = 0>
+    explicit PoissonOpenBC (Geometry const& geom,
+                            IndexType ixtype = IndexType::TheCellType(),
+                            IntVect const& ngrow = IntVect(0));
+
+    void solve (MF& soln, MF const& rhs);
+
+    void define_doit (); // has to be public for cuda
+
+private:
+    Geometry m_geom;
+    Box m_grown_domain;
+    IntVect m_ngrow;
+    OpenBCSolver<typename MF::value_type> m_solver;
+};
+#endif
+
 /**
  * \brief 3D Poisson solver for periodic boundaries in the first two
  * dimensions and Neumann in the last dimension.
@@ -123,6 +150,58 @@ void Poisson<MF>::solve (MF& soln, MF const& rhs)
     });
 }
 
+#if (AMREX_SPACEDIM == 3)
+
+template <typename MF>
+template <typename FA, std::enable_if_t<IsFabArray_v<FA>,int> FOO>
+PoissonOpenBC<MF>::PoissonOpenBC (Geometry const& geom, IndexType ixtype,
+                                  IntVect const& ngrow)
+    : m_geom(geom),
+      m_grown_domain(amrex::grow(amrex::convert(geom.Domain(),ixtype),ngrow)),
+      m_ngrow(ngrow),
+      m_solver(m_grown_domain)
+{
+    define_doit();
+}
+
+template <typename MF>
+void PoissonOpenBC<MF>::define_doit ()
+{
+    using T = typename MF::value_type;
+    auto const& lo = m_grown_domain.smallEnd();
+    auto const dx = T(m_geom.CellSize(0));
+    auto const dy = T(m_geom.CellSize(1));
+    auto const dz = T(m_geom.CellSize(2));
+    auto const gfac = T(1)/T(std::sqrt(T(12)));
+    // 0.125 comes from that there are 8 Gauss quadrature points
+    auto const fac = T(-0.125) * (dx*dy*dz) / (T(4)*Math::pi<T>());
+    m_solver.setGreensFunction([=] AMREX_GPU_DEVICE (int i, int j, int k) -> T
+    {
+        auto x = (T(i-lo[0]) - gfac) * dx; // first Gauss quadrature point
+        auto y = (T(j-lo[1]) - gfac) * dy;
+        auto z = (T(k-lo[2]) - gfac) * dz;
+        T r = 0;
+        for (int gx = 0; gx < 2; ++gx) {
+        for (int gy = 0; gy < 2; ++gy) {
+        for (int gz = 0; gz < 2; ++gz) {
+            auto xg = x + 2*gx*gfac*dx;
+            auto yg = y + 2*gy*gfac*dy;
+            auto zg = z + 2*gz*gfac*dz;
+            r += T(1)/std::sqrt(xg*xg+yg*yg+zg*zg);
+        }}}
+        return fac * r;
+    });
+}
+
+template <typename MF>
+void PoissonOpenBC<MF>::solve (MF& soln, MF const& rhs)
+{
+    AMREX_ASSERT(m_grown_domain.ixType() == soln.ixType() && m_grown_domain.ixType() == rhs.ixType());
+    m_solver.solve(soln, rhs);
+}
+
+#endif /* AMREX_SPACEDIM == 3 */
+
 template <typename MF>
 void PoissonHybrid<MF>::solve (MF& soln, MF const& rhs)
 {