python-ffc 1.6.0-2 / usr / lib / python2.7 / dist-packages / ffc / evaluatebasisderivatives.py

"""Code generation for evaluation of derivatives of finite element
basis values.  This module generates code which is more or less a C++
representation of the code found in FIAT_NEW."""

# Copyright (C) 2007-2013 Kristian B. Oelgaard
#
# This file is part of FFC.
#
# FFC is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# FFC is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with FFC. If not, see <http://www.gnu.org/licenses/>.
#
# Modified by Anders Logg 2013
# Modified by Lizao Li 2015
#
# First added:  2007-04-16
# Last changed: 2015-03-28

# Python modules
import math
import numpy

# FFC modules
from ffc.log import error, ffc_assert
from ffc.evaluatebasis import _compute_basisvalues, _tabulate_coefficients
from ffc.cpp import remove_unused, indent, format

def _evaluate_basis_derivatives_all(data):
    """Like evaluate_basis, but return the values of all basis
    functions (dofs)."""

    if isinstance(data, str):
        return format["exception"]("evaluate_basis_derivatives_all: %s" % data)

    # Prefetch formats.
    f_r, f_s      = format["free indices"][:2]
    f_assign      = format["assign"]
    f_loop        = format["generate loop"]
    f_array       = format["dynamic array"]
    f_dof_vals    = format["dof values"]
    f_comment     = format["comment"]
    f_derivs      = format["call basis_derivatives"]
    f_values      = format["argument values"]
    f_int         = format["int"]
    f_num_derivs  = format["num derivatives"]
    f_double      = format["float declaration"]
    f_component   = format["component"]
    f_mul         = format["mul"]
    f_float       = format["floating point"]
    f_index       = format["matrix index"]
    f_del_array   = format["delete dynamic array"]

    # Initialise return code
    code = []

    # FIXME: KBO: Figure out which return format to use, either:
    # [dN0[0]/dx, dN0[0]/dy, dN0[1]/dx, dN0[1]/dy, dN1[0]/dx,
    # dN1[0]/dy, dN1[1]/dx, dN1[1]/dy, ...]
    # or
    # [dN0[0]/dx, dN1[0]/dx, ..., dN0[1]/dx, dN1[1]/dx, ...,
    # dN0[0]/dy, dN1[0]/dy, ..., dN0[1]/dy, dN1[1]/dy, ...]
    # or
    # [dN0[0]/dx, dN0[1]/dx, ..., dN1[0]/dx, dN1[1]/dx, ...,
    # dN0[0]/dy, dN0[1]/dy, ..., dN1[0]/dy, dN1[1]/dy, ...]
    # for vector (tensor elements), currently returning option 1.

    # FIXME: KBO: For now, just call evaluate_basis_derivatives and
    # map values accordingly, this will keep the amount of code at a
    # minimum. If it turns out that speed is an issue (overhead from
    # calling evaluate_basis), we can easily generate all the code.

    # Get total value shape and space dimension for entire element
    # (possibly mixed).
    physical_value_size = data["physical_value_size"]
    space_dimension = data["space_dimension"]
    max_degree = data["max_degree"]

    # Special case where space dimension is one (constant elements).
    if space_dimension == 1:
        code += [f_comment("Element is constant, calling evaluate_basis_derivatives.")]
        code += [f_derivs(f_int(0), f_values)]
        return "\n".join(code)

    # Compute number of derivatives.
    if data["topological_dimension"]==data["geometric_dimension"]:
        _g = ""
    else:
        _g = "_g"

    # If n == 0, call evaluate_basis.
    code += [format["comment"]("Call evaluate_basis_all if order of derivatives is equal to zero.")]
    cond = format["argument derivative order"] + format["is equal"] + format["int"](0)
    val = [format["call basis_all"]]
    val += [format["return"]("")]
    code += [format["if"](cond, indent("\n".join(val),2))]

    code += _compute_num_derivatives(data["geometric_dimension"], _g)

    if (physical_value_size == 1):
        num_vals = f_num_derivs(_g)
    else:
        num_vals = f_mul([f_int(physical_value_size), f_num_derivs(_g)])

    # Reset values.
    code += ["", f_comment("Set values equal to zero.")]
    name    = f_component(f_values, f_index(f_r, f_s, num_vals))
    lines_s = [f_assign(name, f_float(0.0))]
    loop_s  = [(f_s, 0, num_vals)]
    lines_r = f_loop(lines_s, loop_s)
    loop_r  = [(f_r, 0, space_dimension)]
    code    += f_loop(lines_r, loop_r)

    # If n > max_degree, return zeros.
    code += ["", format["comment"]("If order of derivatives is greater than the maximum polynomial degree, return zeros.")]
    cond = format["argument derivative order"] + format["greater than"] + f_int(max_degree)
    val  = format["return"]("")
    code += [format["if"](cond, indent(val,2))]

    # Declare helper value to hold single dof values and reset.
    code += [f_comment("Helper variable to hold values of a single dof.")]
    nds = data["geometric_dimension"]**max_degree*physical_value_size
    code += [format["declaration"](f_double, f_component(f_dof_vals, f_int(nds)))]
    line  = [f_assign(f_component(f_dof_vals, f_r), f_float(0.0))]
    code += f_loop(line, [(f_r, 0, nds)])

    # Create loop over dofs that calls evaluate_basis_derivatives for a single dof and
    # inserts the values into the global array.
    code += ["", f_comment("Loop dofs and call evaluate_basis_derivatives.")]
    name  = f_component(f_values, f_index(f_r, f_s, num_vals))
    value = f_component(f_dof_vals, f_s)
    lines_s  = [f_assign(name, value)]
    loop_s   = [(f_s, 0, num_vals)]

    lines_r  = [f_derivs(f_r, f_dof_vals)]
    lines_r += f_loop(lines_s, loop_s)
    loop_r   = [(f_r, 0, space_dimension)]
    code    += f_loop(lines_r, loop_r)

    # Generate bode (no need to remove unused).
    return "\n".join(code)

def _evaluate_basis_derivatives(data):
    """Evaluate the derivatives of an element basisfunction at a point. The values are
    computed as in FIAT as the matrix product of the coefficients (computed at compile time),
    basisvalues which are dependent on the coordinate and thus have to be computed at
    run time and combinations (depending on the order of derivative) of dmats
    tables which hold the derivatives of the expansion coefficients."""

    if isinstance(data, str):
        return format["exception"]("evaluate_basis_derivatives: %s" % data)

    # Initialise return code.
    code = []

    # Get the element cell domain, geometric and topological dimension.
    element_cellname = data["cellname"]
    gdim = data["geometric_dimension"]
    tdim = data["topological_dimension"]
    max_degree = data["max_degree"]

    # Compute number of derivatives that has to be computed, and
    # declare an array to hold the values of the derivatives on the
    # reference element.
    code += [""]
    if tdim == gdim:
        _t = ""
        _g = ""
        code += _compute_num_derivatives(tdim, "")

        # Reset all values.
        code += _reset_values(data, _g)

        # Handle values of argument 'n'.
        code += _handle_degree(max_degree)

        # If max_degree is zero, return code (to avoid declarations such as
        # combinations[1][0]) and because there's nothing to compute.)
        if max_degree == 0:
            return remove_unused("\n".join(code))

        # Generate geo code.
        code += _geometry_related_code(data, tdim, gdim, element_cellname)

        # Generate all possible combinations of derivatives.
        code += _generate_combinations(tdim, "", max_degree)
    else:
        _t = "_t"
        _g = "_g"
        code += _compute_num_derivatives(tdim, _t)
        code += [""]
        code += _compute_num_derivatives(gdim, _g)

        # Reset all values.
        code += _reset_values(data, _g)

        # Handle values of argument 'n'.
        code += _handle_degree(max_degree)

        # If max_degree is zero, return code (to avoid declarations such as
        # combinations[1][0]) and because there's nothing to compute.)
        if max_degree == 0:
            return remove_unused("\n".join(code))

        # Generate geo code.
        code += _geometry_related_code(data, tdim, gdim, element_cellname)

        # Generate all possible combinations of derivatives.
        code += _generate_combinations(tdim, _t, max_degree)
        code += _generate_combinations(gdim, _g, max_degree)

    # Generate the transformation matrix.
    code += _generate_transform(element_cellname, gdim, tdim, max_degree)

    # Create code for all basis values (dofs).
    dof_cases = []
    for dof in data["dof_data"]:
        dof_cases.append(_generate_dof_code(data, dof))
    code += [format["switch"](format["argument basis num"], dof_cases)]
    code = remove_unused("\n".join(code))
    #code = "\n".join(code)
    return code

def _handle_degree(max_degree):
    """Check value of argument 'n' against the maximum polynomial degree of the
    finite element. If user ask for n>max_degree return an appropriate number
    of zeros in the 'values' array. If n==0, simply direct call to
    evaluate_basis."""

    code = []

    # If n == 0, call evaluate_basis.
    code += [format["comment"]("Call evaluate_basis if order of derivatives is equal to zero.")]
    cond = format["argument derivative order"] + format["is equal"] + format["int"](0)
    val = [format["call basis"](format["argument dof num"], format["argument values"])]
    val += [format["return"]("")]
    code += [format["if"](cond, indent("\n".join(val),2))]

    # If n > max_degree, derivatives are always zero. Since the appropriate number of
    # zeros have already been inserted into the 'values' array simply return.
    code += [format["comment"]("If order of derivatives is greater than the maximum polynomial degree, return zeros.")]
    cond = format["argument derivative order"] + format["greater than"] + format["int"](max_degree)
    val = format["return"]("")
    code += [format["if"](cond, val)]

    return code

def _geometry_related_code(data, tdim, gdim, element_cellname):
    code = []
    # Get code snippets for Jacobian, inverse of Jacobian and mapping of
    # coordinates from physical element to the FIAT reference element.
    code += [format["compute_jacobian"](tdim, gdim)]
    code += [format["compute_jacobian_inverse"](tdim, gdim)]
    if data["needs_oriented"]:
        code += [format["orientation"](tdim, gdim)]
    code += ["", format["fiat coordinate map"](element_cellname, gdim)]
    return code

def _compute_num_derivatives(dimension, suffix=""):
    """Computes the number of derivatives of order 'n' as dimension()^n.

    Dimension will be the element topological dimension for the number
    of derivatives in local coordinates, and the geometric dimension
    for the number of derivatives in phyisical coordinates.
    """
    # Prefetch formats.
    f_int         = format["int"]
    f_num_derivs  = format["num derivatives"](suffix)

    # Use loop to compute power since using std::pow() result in an
    # ambiguous call.
    code = [format["comment"]("Compute number of derivatives.")]
    code.append(format["declaration"](format["uint declaration"],
                                      f_num_derivs, f_int(1)))
    loop_vars = [(format["free indices"][0], 0,
                  format["argument derivative order"])]
    lines = [format["imul"](f_num_derivs, f_int(dimension))]
    code += format["generate loop"](lines, loop_vars)

    return code

def _generate_combinations(dimension, suffix, max_degree):
    "Generate all possible combinations of derivatives of order 'n'."

    nds = dimension**max_degree

    # Use code from format.
    code = ["", format["combinations"]\
            % {"combinations": format["derivative combinations"](suffix),\
               "dimension-1": dimension-1,\
               "num_derivatives" : format["num derivatives"](suffix),\
               "n": format["argument derivative order"],
               "max_num_derivatives":format["int"](nds),
               "max_degree":format["int"](max_degree)}]
    return code

def _generate_transform(element_cellname, gdim, tdim, max_degree):
    """Generate the transformation matrix, which is used to transform
    derivatives from reference element back to the physical element."""

    max_g_d = gdim**max_degree
    max_t_d = tdim**max_degree
    # Generate code to construct the inverse of the Jacobian
    if (element_cellname in ["interval", "triangle", "tetrahedron"]):
        code = ["", format["transform snippet"][element_cellname][gdim]\
        % {"transform": format["transform matrix"],\
           "num_derivatives" : format["num derivatives"](""),\
           "n": format["argument derivative order"],\
           "combinations": format["derivative combinations"](""),\
           "K":format["transform Jinv"],
           "max_g_deriv":max_g_d, "max_t_deriv":max_t_d}]
    else:
        error("Cannot generate transform for shape: %s" % element_cellname)

    return code

def _reset_values(data, suffix):
    "Reset all components of the 'values' array as it is a pointer to an array."

    # Prefetch formats.
    f_assign  = format["assign"]
    f_r       = format["free indices"][0]

    code = ["", format["comment"]("Reset values. Assuming that values is always an array.")]

    # Get value shape and reset values. This should also work for TensorElement,
    # scalar are empty tuples, therefore (1,) in which case value_shape = 1.
    physical_value_size = data["physical_value_size"]

    # Only multiply by value shape if different from 1.
    if physical_value_size == 1:
        num_vals = format["num derivatives"](suffix)
    else:
        num_vals = format["mul"]([format["int"](physical_value_size), format["num derivatives"](suffix)])
    name = format["component"](format["argument values"], f_r)
    loop_vars = [(f_r, 0, num_vals)]
    lines = [f_assign(name, format["floating point"](0))]
    code += format["generate loop"](lines, loop_vars)

    return code + [""]

def _generate_dof_code(data, dof_data):
    "Generate code for a basis."

    code = []

    # Compute basisvalues, from evaluatebasis.py.
    code += _compute_basisvalues(data, dof_data)

    # Tabulate coefficients.
    code += _tabulate_coefficients(dof_data)

    # Tabulate coefficients for derivatives.
    code += _tabulate_dmats(dof_data)

    # Compute the derivatives of the basisfunctions on the reference (FIAT) element,
    # as the dot product of the new coefficients and basisvalues.
    code += _compute_reference_derivatives(data, dof_data)

    # Transform derivatives to physical element by multiplication with the transformation matrix.
    code += _transform_derivatives(data, dof_data)

    code = remove_unused("\n".join(code))

    return code

def _tabulate_dmats(dof_data):
    "Tabulate the derivatives of the polynomial base"

    code = []

    # Prefetch formats to speed up code generation.
    f_table     = format["static const float declaration"]
    f_dmats     = format["dmats"]
    f_component = format["component"]
    f_decl      = format["declaration"]
    f_tensor    = format["tabulate tensor"]
    f_new_line  = format["new line"]

    # Get derivative matrices (coefficients) of basis functions, computed by FIAT at compile time.
    derivative_matrices = dof_data["dmats"]

    code += [format["comment"]("Tables of derivatives of the polynomial base (transpose).")]

    # Generate tables for each spatial direction.
    for i, dmat in enumerate(derivative_matrices):

        # Extract derivatives for current direction (take transpose, FIAT_NEW PolynomialSet.tabulate()).
        matrix = numpy.transpose(dmat)

        # Get shape and check dimension (This is probably not needed).
        shape = numpy.shape(matrix)
        ffc_assert(shape[0] == shape[1] == dof_data["num_expansion_members"], "Something is wrong with the shape of dmats.")

        # Declare varable name for coefficients.
        name = f_component(f_dmats(i), [shape[0], shape[1]])
        code += [f_decl(f_table, name, f_new_line + f_tensor(matrix)), ""]

    return code

def _reset_dmats(shape_dmats, indices):
    "Set values in dmats equal to the identity matrix."
    f_assign  = format["assign"]
    f_float   = format["floating point"]
    i,j = indices

    code = [format["comment"]("Resetting dmats values to compute next derivative.")]
    dmats_old = format["component"](format["dmats"](""), [i, j])
    lines = [f_assign(dmats_old, f_float(0.0))]
    lines += [format["if"](i + format["is equal"] + j,\
              f_assign(dmats_old, f_float(1.0)))]
    loop_vars = [(i, 0, shape_dmats[0]), (j, 0, shape_dmats[1])]
    code += format["generate loop"](lines, loop_vars)
    return code

def _update_dmats(shape_dmats, indices):
    "Update values in dmats_old with values in dmats and set values in dmats to zero."
    f_assign    = format["assign"]
    f_component = format["component"]
    i,j = indices

    code = [format["comment"]("Updating dmats_old with new values and resetting dmats.")]
    dmats = f_component(format["dmats"](""), [i, j])
    dmats_old = f_component(format["dmats old"], [i, j])
    lines = [f_assign(dmats_old, dmats), f_assign(dmats, format["floating point"](0.0))]
    loop_vars = [(i, 0, shape_dmats[0]), (j, 0, shape_dmats[1])]
    code += format["generate loop"](lines, loop_vars)
    return code

def _compute_dmats(num_dmats, shape_dmats, available_indices, deriv_index, _t):
    "Compute values of dmats as a matrix product."
    f_comment = format["comment"]
    s, t, u = available_indices

    # Reset dmats_old
    code = _reset_dmats(shape_dmats, [t, u])
    code += ["", f_comment("Looping derivative order to generate dmats.")]

    # Set dmats matrix equal to dmats_old
    lines = _update_dmats(shape_dmats, [t, u])

    lines += ["", f_comment("Update dmats using an inner product.")]

    # Create dmats matrix by multiplication
    comb = format["component"](format["derivative combinations"](_t), [deriv_index, s])
    for i in range(num_dmats):
        lines += _dmats_product(shape_dmats, comb, i, [t, u])

    loop_vars = [(s, 0, format["argument derivative order"])]
    code += format["generate loop"](lines, loop_vars)

    return code

def _dmats_product(shape_dmats, index, i, indices):
    "Create product to update dmats."
    f_loop      = format["generate loop"]
    f_component = format["component"]
    t, u = indices
    tu = t + u

    dmats = f_component(format["dmats"](""), [t, u])
    dmats_old = f_component(format["dmats old"], [tu, u])
    value = format["multiply"]([f_component(format["dmats"](i), [t, tu]), dmats_old])
    name = format["iadd"](dmats, value)
    lines = f_loop([name], [(tu, 0, shape_dmats[0])])
    loop_vars = [(t, 0, shape_dmats[0]), (u, 0, shape_dmats[1])]
    code = [format["if"](index + format["is equal"] + str(i),\
            "\n".join(f_loop(lines, loop_vars)))]
    return code

def _compute_reference_derivatives(data, dof_data):
    """Compute derivatives on the reference element by recursively multiply coefficients with
    the relevant derivatives of the polynomial base until the requested order of derivatives
    has been reached. After this take the dot product with the basisvalues."""

    # Prefetch formats to speed up code generation
    f_comment       = format["comment"]
    f_num_derivs    = format["num derivatives"]
    f_mul           = format["mul"]
    f_int           = format["int"]
    f_matrix_index  = format["matrix index"]
    f_coefficients  = format["coefficients"]
#    f_dof           = format["local dof"]
    f_basisvalues   = format["basisvalues"]
    f_const_double  = format["const float declaration"]
    f_group         = format["grouping"]
    f_transform     = format["transform"]
    f_double        = format["float declaration"]
    f_component     = format["component"]
    f_tmp           = format["tmp ref value"]
    f_dmats         = format["dmats"]
    f_dmats_old     = format["dmats old"]
    f_assign        = format["assign"]
    f_decl          = format["declaration"]
    f_iadd          = format["iadd"]
    f_add           = format["add"]
    f_tensor        = format["tabulate tensor"]
    f_new_line      = format["new line"]
    f_loop          = format["generate loop"]
    f_derivatives   = format["reference derivatives"]
    f_array         = format["dynamic array"]
    f_float         = format["floating point"]
    f_inv           = format["inverse"]
    f_detJ          = format["det(J)"]
    f_inner         = format["inner product"]

    f_r, f_s, f_t, f_u = format["free indices"]

    tdim = data["topological_dimension"]
    gdim = data["geometric_dimension"]
    max_degree = data["max_degree"]

    if tdim == gdim:
        _t = ""
        _g = ""
    else:
        _t = "_t"
        _g = "_g"

    # Get number of components.
    num_components = dof_data["num_components"]

    # Get shape of derivative matrix (they should all have the same shape) and
    # verify that it is a square matrix.
    shape_dmats = numpy.shape(dof_data["dmats"][0])
    ffc_assert(shape_dmats[0] == shape_dmats[1],\
               "Something is wrong with the dmats:\n%s" % str(dof_data["dmats"]))

    code = [f_comment("Compute reference derivatives.")]

    # Declare pointer to array that holds derivatives on the FIAT element
    code += [f_comment("Declare array of derivatives on FIAT element.")]
    # The size of the array of reference derivatives is equal to the number of derivatives
    # times the number of components of the basis element
    if (num_components == 1):
        num_vals = f_num_derivs(_t)
    else:
        num_vals = f_mul([f_int(num_components), f_num_derivs(_t)])

    nds = tdim**max_degree*num_components
    code += [format["declaration"](f_double, f_component(f_derivatives, f_int(nds)))]
    line  = [f_assign(f_component(f_derivatives, f_r), f_float(0.0))]
    code += f_loop(line, [(f_r, 0, nds)])
    code += [""]

    mapping = dof_data["mapping"]
    if "piola" in mapping:
        # In either of the Piola cases, the value space of the derivatives is the geometric dimension rather than the topological dimension.
        code += [f_comment("Declare array of reference derivatives on physical element.")]

        _p = "_p"
        num_components_p = gdim

        nds = tdim**max_degree*gdim
        code += [format["declaration"](f_double, f_component(f_derivatives+_p, f_int(nds)))]
        line  = [f_assign(f_component(f_derivatives+_p, f_r), f_float(0.0))]
        code += f_loop(line, [(f_r, 0, nds)])
        code += [""]
    else:
        _p = ""
        num_components_p = num_components

    # Declare matrix of dmats (which will hold the matrix product of all combinations)
    # and dmats_old which is needed in order to perform the matrix product.
    value = f_tensor(numpy.eye(shape_dmats[0]))
    code += [f_comment("Declare derivative matrix (of polynomial basis).")]
    name = f_component(f_dmats(""), [shape_dmats[0], shape_dmats[1]])
    code += [f_decl(f_double, name, f_new_line + value), ""]
    code += [f_comment("Declare (auxiliary) derivative matrix (of polynomial basis).")]
    name = f_component(f_dmats_old, [shape_dmats[0], shape_dmats[1]])
    code += [f_decl(f_double, name, f_new_line + value), ""]

    # Compute dmats as a recursive matrix product
    lines = _compute_dmats(len(dof_data["dmats"]), shape_dmats, [f_s, f_t, f_u], f_r, _t)

    # Compute derivatives for all components
    lines_c = []
    for i in range(num_components):
        name = f_component(f_derivatives, f_matrix_index(i, f_r, f_num_derivs(_t)))
        coeffs = f_component(f_coefficients(i), f_s)
        dmats = f_component(f_dmats(""), [f_s, f_t])
        basis = f_component(f_basisvalues, f_t)
        lines_c.append(f_iadd(name, f_mul([coeffs, dmats, basis])))
    loop_vars_c = [(f_s, 0, shape_dmats[0]),(f_t, 0, shape_dmats[1])]
    lines += f_loop(lines_c, loop_vars_c)

    # Apply transformation if applicable.
    if mapping == "affine":
        pass
    elif mapping == "contravariant piola":
        lines += ["", f_comment\
                ("Using contravariant Piola transform to map values back to the physical element.")]
        # Get temporary values before mapping.
        lines += [f_const_double(f_tmp(i),\
                  f_component(f_derivatives, f_matrix_index(i, f_r, f_num_derivs(_t)))) for i in range(num_components)]

        # Create names for inner product.
        basis_col = [f_tmp(j) for j in range(tdim)]
        for i in range(num_components_p):
            # Create Jacobian.
            jacobian_row = [f_transform("J", i, j, gdim, tdim, None) for j in range(tdim)]

            # Create inner product and multiply by inverse of Jacobian.
            inner = [f_mul([jacobian_row[j], basis_col[j]]) for j in range(tdim)]
            sum_ = f_group(f_add(inner))
            value = f_mul([f_inv(f_detJ(None)), sum_])
            name = f_component(f_derivatives+_p, f_matrix_index(i, f_r, f_num_derivs(_t)))
            lines += [f_assign(name, value)]
    elif mapping == "covariant piola":
        lines += ["", f_comment\
                ("Using covariant Piola transform to map values back to the physical element")]
        # Get temporary values before mapping.
        lines += [f_const_double(f_tmp(i),\
                  f_component(f_derivatives, f_matrix_index(i, f_r, f_num_derivs(_t)))) for i in range(num_components)]
        # Create names for inner product.
        basis_col = [f_tmp(j) for j in range(tdim)]
        for i in range(num_components_p):
            # Create inverse of Jacobian.
            inv_jacobian_column = [f_transform("JINV", j, i, tdim, gdim, None) for j in range(tdim)]

            # Create inner product of basis and inverse of Jacobian.
            inner = [f_mul([inv_jacobian_column[j], basis_col[j]]) for j in range(tdim)]
            value = f_group(f_add(inner))
            name = f_component(f_derivatives+_p, f_matrix_index(i, f_r, f_num_derivs(_t)))
            lines += [f_assign(name, value)]
    elif mapping == "pullback as metric":
        lines += ["", f_comment("Using metric pullback to map values back to the physical element")]
        lines += [f_const_double(f_tmp(i),
                                f_component(f_derivatives,
                                            f_matrix_index(i, f_r, f_num_derivs(_t))))
                  for i in range(num_components)]
        basis_col = [f_tmp(j) for j in range(num_components)]
        for p in range(num_components):
            # unflatten the indices
            i = p // tdim
            l = p % tdim
            # g_il = K_ji G_jk K_kl
            value = f_group(f_inner(
                [f_inner([f_transform("JINV", j, i, tdim, gdim, None)
                          for j in range(tdim)],
                         [basis_col[j * tdim + k] for j in range(tdim)])
                 for k in range(tdim)],
                [f_transform("JINV", k, l, tdim, gdim, None)
                 for k in range(tdim)]))
            name = f_component(f_derivatives+_p, f_matrix_index(p, f_r, f_num_derivs(_t)))
            lines += [f_assign(name, value)]
    else:
        error("Unknown mapping: %s" % mapping)

    # Generate loop over number of derivatives.
    # Loop all derivatives and compute value of the derivative as:
    # deriv_on_ref[r] = coeff[dof][s]*dmat[s][t]*basis[t]
    code += [f_comment("Loop possible derivatives.")]
    loop_vars = [(f_r, 0, f_num_derivs(_t))]
    code += f_loop(lines, loop_vars)

    return code + [""]

def _transform_derivatives(data, dof_data):
    """Transform derivatives back to the physical element by applying the
    transformation matrix."""

    # Prefetch formats to speed up code generation.
    f_loop        = format["generate loop"]
    f_num_derivs  = format["num derivatives"]
    f_derivatives = format["reference derivatives"]
    f_values      = format["argument values"]
    f_mul         = format["mul"]
    f_iadd        = format["iadd"]
    f_component   = format["component"]
    f_transform   = format["transform matrix"]
    f_r, f_s      = format["free indices"][:2]
    f_index       = format["matrix index"]

    if data["topological_dimension"]==data["geometric_dimension"]:
        _t = ""
        _g = ""
    else:
        _t = "_t"
        _g = "_g"

    # Get number of components and offset.
    num_components = dof_data["num_components"]
    offset = dof_data["offset"]

    mapping = dof_data["mapping"]
    if "piola" in mapping:
        # In either of the Piola cases, the value space of the derivatives is the geometric dimension rather than the topological dimension.
        _p = "_p"
        num_components_p = data["geometric_dimension"]
    else:
        _p = ""
        num_components_p = num_components

    code = [format["comment"]("Transform derivatives back to physical element")]

    lines = []
    for i in range(num_components_p):
        access_name = f_index(offset + i, f_r, f_num_derivs(_g))
        name = f_component(f_values, access_name)
        access_val = f_index(i, f_s, f_num_derivs(_t))
        value = f_mul([f_component(f_transform, [f_r, f_s]), f_component(f_derivatives+_p, access_val)])
        lines += [f_iadd(name, value)]

    loop_vars = [(f_r, 0, f_num_derivs(_g)), (f_s, 0, f_num_derivs(_t))]
    code += f_loop(lines, loop_vars)
    return code