Repository: mscroggs/symfem

Issue:
Fix rHCT element
**Describe the bug**
The rHCT element seems to be wrong once integrated. I noticed it when I tried to build the stiffness matrix for the Poisson equation using these elements.
It is clear when one tries to integrate it using symfem or sympy.

**To Reproduce**
Minimal code to reproduce the behavior:
```python
import symfem
from symfem.symbols import x
from sympy import Symbol, integrate, Rational

#Compute integral using symfem
element = symfem.create_element("triangle", "rHCT", 3)
ref = element.reference
f1 = element.get_basis_function(1).directional_derivative((1,0))
f2 = element.get_basis_function(6).directional_derivative((1,0))
integrand = f1*f2
#f1*f2 is 0 everywhere, except on triangle ((0, 1), (0, 0), (1/3, 1/3))
#where it equals (42*x**2 - 24*x*y - 6*x)*(18*x**2 + 6*x*y - 10*x - y + 1)
print(integrand.integral(ref, x)) #=0

#Compute it using sympy
z = Symbol("x")
y = Symbol("y")
expr = (42*z**2 - 24*z*y - 6*z)*(18*z**2 + 6*z*y - 10*z - y + 1) #Copied from what is given by symfem
int_expr = integrate(expr, (y, z, 1-2*z)) #parametrization of triangle ((0, 1), (0, 0), (1/3, 1/3)) : x<y<1-2*x
print(integrate(int_expr, (z,0, Rational(1,3))))#=1/108
```
**Expected behavior**
The integral should be correct in this example

**Screenshots**
Not applicable.

**Additional context**
No other context needed (I guess?)


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Patch:
diff --git a/CHANGELOG_SINCE_LAST_VERSION.md b/CHANGELOG_SINCE_LAST_VERSION.md
index ab4a35a..80efcc1 100644
--- a/CHANGELOG_SINCE_LAST_VERSION.md
+++ b/CHANGELOG_SINCE_LAST_VERSION.md
@@ -1,3 +1,4 @@
 - Added enriched Galerkin element
 - Improved plotting
 - Added enriched vector Galerkin element
+- Fixed bug in integration of piecewise functions
diff --git a/symfem/basis_functions.py b/symfem/basis_functions.py
index 1c7a053..6c0a7c5 100644
--- a/symfem/basis_functions.py
+++ b/symfem/basis_functions.py
@@ -10,7 +10,7 @@ import sympy
 from .functions import AnyFunction, FunctionInput, ScalarFunction, SympyFormat, ValuesToSubstitute
 from .geometry import PointType
 from .references import Reference
-from .symbols import AxisVariables, AxisVariablesNotSingle, t
+from .symbols import AxisVariables, AxisVariablesNotSingle, t, x
 
 
 class BasisFunction(AnyFunction):
@@ -282,17 +282,21 @@ class BasisFunction(AnyFunction):
         """
         raise self.get_function().norm()
 
-    def integral(self, domain: Reference, vars: AxisVariablesNotSingle = t) -> AnyFunction:
+    def integral(
+        self, domain: Reference, vars: AxisVariablesNotSingle = x,
+        dummy_vars: AxisVariablesNotSingle = t
+    ) -> ScalarFunction:
         """Compute the integral of the function.
 
         Args:
-            domain: The domain to integrate over
-            vars: The variables to integrate over
+            domain: The domain of the integral
+            vars: The variables to integrate with respect to
+            dummy_vars: The dummy variables to use inside the integral
 
         Returns:
             The integral
         """
-        return self.get_function().integral(domain, vars)
+        return self.get_function().integral(domain, vars, dummy_vars)
 
     def subs(self, vars: AxisVariables, values: ValuesToSubstitute) -> BasisFunction:
         """Substitute values into the function.
diff --git a/symfem/functions.py b/symfem/functions.py
index d7f45ef..209b4b4 100644
--- a/symfem/functions.py
+++ b/symfem/functions.py
@@ -314,12 +314,16 @@ class AnyFunction(ABC):
         pass
 
     @abstractmethod
-    def integral(self, domain: Reference, vars: AxisVariablesNotSingle = t):
+    def integral(
+        self, domain: Reference, vars: AxisVariablesNotSingle = x,
+        dummy_vars: AxisVariablesNotSingle = t
+    ) -> ScalarFunction:
         """Compute the integral of the function.
 
         Args:
             domain: The domain of the integral
             vars: The variables to integrate with respect to
+            dummy_vars: The dummy variables to use inside the integral
 
         Returns:
             The integral
@@ -758,22 +762,27 @@ class ScalarFunction(AnyFunction):
         """
         return ScalarFunction(abs(self._f))
 
-    def integral(self, domain: Reference, vars: AxisVariablesNotSingle = t) -> ScalarFunction:
+    def integral(
+        self, domain: Reference, vars: AxisVariablesNotSingle = x,
+        dummy_vars: AxisVariablesNotSingle = t
+    ) -> ScalarFunction:
         """Compute the integral of the function.
 
         Args:
             domain: The domain of the integral
             vars: The variables to integrate with respect to
+            dummy_vars: The dummy variables to use inside the integral
 
         Returns:
             The integral
         """
-        limits = domain.integration_limits(vars)
-
+        limits = domain.integration_limits(dummy_vars)
         point = VectorFunction(domain.origin)
-        for ti, a in zip(t, domain.axes):
+        for ti, a in zip(dummy_vars, domain.axes):
             point += ti * VectorFunction(a)
-        out = self._f.subs(x, point)
+        out = self._f * 1
+        for v, p in zip(vars, point):
+            out = out.subs(v, p)
 
         if len(limits[0]) == 2:
             for i in limits:
@@ -1159,12 +1168,16 @@ class VectorFunction(AnyFunction):
             a += i._f ** 2
         return ScalarFunction(sympy.sqrt(a))
 
-    def integral(self, domain: Reference, vars: AxisVariablesNotSingle = t):
+    def integral(
+        self, domain: Reference, vars: AxisVariablesNotSingle = x,
+        dummy_vars: AxisVariablesNotSingle = t
+    ) -> ScalarFunction:
         """Compute the integral of the function.
 
         Args:
             domain: The domain of the integral
             vars: The variables to integrate with respect to
+            dummy_vars: The dummy variables to use inside the integral
 
         Returns:
             The integral
@@ -1587,12 +1600,16 @@ class MatrixFunction(AnyFunction):
         """
         raise NotImplementedError()
 
-    def integral(self, domain: Reference, vars: AxisVariablesNotSingle = t):
+    def integral(
+        self, domain: Reference, vars: AxisVariablesNotSingle = x,
+        dummy_vars: AxisVariablesNotSingle = t
+    ) -> ScalarFunction:
         """Compute the integral of the function.
 
         Args:
             domain: The domain of the integral
             vars: The variables to integrate with respect to
+            dummy_vars: The dummy variables to use inside the integral
 
         Returns:
             The integral
diff --git a/symfem/piecewise_functions.py b/symfem/piecewise_functions.py
index c714e04..ac8ddb2 100644
--- a/symfem/piecewise_functions.py
+++ b/symfem/piecewise_functions.py
@@ -6,8 +6,8 @@ import typing
 
 import sympy
 
-from .functions import (AnyFunction, FunctionInput, SympyFormat, ValuesToSubstitute, VectorFunction,
-                        _to_sympy_format, parse_function_input)
+from .functions import (AnyFunction, FunctionInput, ScalarFunction, SympyFormat, ValuesToSubstitute,
+                        VectorFunction, _to_sympy_format, parse_function_input)
 from .geometry import (PointType, SetOfPoints, SetOfPointsInput, parse_set_of_points_input,
                        point_in_quadrilateral, point_in_tetrahedron, point_in_triangle)
 from .references import Reference
@@ -422,20 +422,27 @@ class PiecewiseFunction(AnyFunction):
         return PiecewiseFunction(
             {shape: f.norm() for shape, f in self._pieces.items()}, self.tdim)
 
-    def integral(self, domain: Reference, vars: AxisVariablesNotSingle = t) -> AnyFunction:
+    def integral(
+        self, domain: Reference, vars: AxisVariablesNotSingle = x,
+        dummy_vars: AxisVariablesNotSingle = t
+    ) -> ScalarFunction:
         """Compute the integral of the function.
 
         Args:
             domain: The domain of the integral
             vars: The variables to integrate with respect to
+            dummy_vars: The dummy variables to use inside the integral
 
         Returns:
             The integral
         """
-        # TODO: Add check that the domain is a subset of one piece
-        # TODO: Add integral over multiple pieces
-        p = self.get_piece(domain.midpoint())
-        return p.integral(domain, vars)
+        result = ScalarFunction(0)
+        for shape, f in self._pieces.items():
+            ref = _piece_reference(self.tdim, shape)
+            sub_domain = ref.intersection(domain)
+            if sub_domain is not None:
+                result += f.integral(sub_domain, vars, dummy_vars)
+        return result
 
     def det(self) -> PiecewiseFunction:
         """Compute the determinant.
@@ -500,25 +507,11 @@ class PiecewiseFunction(AnyFunction):
             value_scale: The scale factor for the function values
             n: The number of points per side for plotting
         """
-        from .create import create_reference
         from .plotting import Picture
         assert isinstance(img, Picture)
 
         for shape, f in self._pieces.items():
-            if self.tdim == 2:
-                if len(shape) == 3:
-                    ref = create_reference("triangle", shape)
-                elif len(shape) == 4:
-                    ref = create_reference("quadrilateral", shape)
-                else:
-                    raise ValueError("Unsupported cell type")
-            elif self.tdim == 3:
-                if len(shape) == 4:
-                    ref = create_reference("tetrahedron", shape)
-                else:
-                    raise ValueError("Unsupported cell type")
-            else:
-                raise ValueError("Unsupported tdim")
+            ref = _piece_reference(self.tdim, shape)
             f.plot_values(ref, img, value_scale, n // 2)
 
     def with_floats(self) -> AnyFunction:
@@ -529,3 +522,22 @@ class PiecewiseFunction(AnyFunction):
         """
         return PiecewiseFunction(
             {shape: f.with_floats() for shape, f in self._pieces.items()}, self.tdim)
+
+
+def _piece_reference(tdim, shape):
+    """Create a reference element for a single piece."""
+    from .create import create_reference
+    if tdim == 2:
+        if len(shape) == 3:
+            return create_reference("triangle", shape)
+        elif len(shape) == 4:
+            return create_reference("quadrilateral", shape)
+        else:
+            raise ValueError("Unsupported cell type")
+    elif tdim == 3:
+        if len(shape) == 4:
+            return create_reference("tetrahedron", shape)
+        else:
+            raise ValueError("Unsupported cell type")
+    else:
+        raise ValueError("Unsupported tdim")
diff --git a/symfem/polynomials.py b/symfem/polynomials.py
index bf6d99b..b6fef54 100644
--- a/symfem/polynomials.py
+++ b/symfem/polynomials.py
@@ -1076,7 +1076,7 @@ def orthonormal_basis(
     ref = create_reference(cell)
     if variables is None:
         variables = x
-    norms = [sympy.sqrt((f ** 2).integral(ref, variables)) for f in poly[0]]
+    norms = [sympy.sqrt((f ** 2).integral(ref, dummy_vars=variables)) for f in poly[0]]
     for i, n in enumerate(norms):
         for j in range(len(poly)):
             poly[j][i] /= n
diff --git a/symfem/references.py b/symfem/references.py
index bfa2a3a..3d70d22 100644
--- a/symfem/references.py
+++ b/symfem/references.py
@@ -17,6 +17,49 @@ IntLimits = typing.List[typing.Union[
     typing.Tuple[sympy.core.symbol.Symbol, sympy.core.expr.Expr]]]
 
 
+def _which_side(vs: SetOfPoints, p: PointType, q: PointType) -> typing.Optional[int]:
+    """Check which side of a line or plane a set of points are.
+
+    Args:
+        vs: The set of points
+        p: A point on the line or plane
+        q: Another point on the line (2D) or the normal to the plane (3D)
+
+    Returns:
+        2 if the points are all to the left, 1 if the points are all to the left or on the line,
+        0 if the points are all on the line, -1 if the points are all to the right or on the line,
+        -1 if the points are all to the right, None if there are some points on either side.
+    """
+    sides = []
+    for v in vs:
+        if len(q) == 2:
+            cross = (v[0] - p[0]) * (q[1] - p[1]) - (v[1] - p[1]) * (q[0] - p[0])
+        elif len(q) == 3:
+            cross = (v[0] - p[0]) * q[0] + (v[1] - p[1]) * q[1] + (v[2] - p[2]) * q[2]
+        else:
+            return None
+        if cross == 0:
+            sides.append(0)
+        elif cross > 0:
+            sides.append(1)
+        else:
+            sides.append(-1)
+
+    if -1 in sides and 1 in sides:
+        return None
+    if 1 in sides:
+        if 0 in sides:
+            return 1
+        else:
+            return 2
+    if -1 in sides:
+        if 0 in sides:
+            return -1
+        else:
+            return -2
+    return 0
+
+
 def _vsub(v: PointTypeInput, w: PointTypeInput) -> PointType:
     """Subtract.
 
@@ -155,6 +198,45 @@ class Reference(ABC):
         """
         return self.vertices
 
+    def intersection(self, other: Reference) -> typing.Optional[Reference]:
+        """Get the intersection of two references.
+
+        Returns:
+            A reference element that is the intersection
+        """
+        if self.gdim != other.gdim:
+            raise ValueError("Incompatible cell dimensions")
+
+        for cell1, cell2 in [(self, other), (other, self)]:
+            try:
+                for v in cell1.vertices:
+                    if not cell2.contains(v):
+                        break
+                else:
+                    return cell1
+            except NotImplementedError:
+                pass
+        for cell1, cell2 in [(self, other), (other, self)]:
+            if cell1.gdim == 2:
+                for e in cell1.edges:
+                    p = cell1.vertices[e[0]]
+                    q = cell1.vertices[e[1]]
+                    dir1 = _which_side(cell1.vertices, p, q)
+                    dir2 = _which_side(cell2.vertices, p, q)
+                    if dir1 is not None and dir2 is not None and dir1 * dir2 < 0:
+                        return None
+            if cell1.gdim == 3:
+                for i in range(cell1.sub_entity_count(2)):
+                    face = cell1.sub_entity(2, i)
+                    p = face.midpoint()
+                    n = face.normal()
+                    dir1 = _which_side(cell1.vertices, p, n)
+                    dir2 = _which_side(cell2.vertices, p, n)
+                    if dir1 is not None and dir2 is not None and dir1 * dir2 < 0:
+                        return None
+
+        raise NotImplementedError("Intersection of these elements is not yet supported")
+
     @abstractmethod
     def default_reference(self) -> Reference:
         """Get the default reference for this cell type.
@@ -1028,9 +1110,17 @@ class Triangle(Reference):
         Returns:
             Is the point contained in the reference?
         """
-        if self.vertices != self.reference_vertices:
-            raise NotImplementedError()
-        return 0 <= point[0] and 0 <= point[1] and sum(point) <= 1
+        if self.vertices == self.reference_vertices:
+            return 0 <= point[0] and 0 <= point[1] and sum(point) <= 1
+        elif self.gdim == 2:
+            po = _vsub(point, self.origin)
+            det = self.axes[0][0] * self.axes[1][1] - self.axes[0][1] * self.axes[1][0]
+            t0 = (self.axes[1][1] * po[0] - self.axes[1][0] * po[1]) / det
+            t1 = (self.axes[0][0] * po[1] - self.axes[0][1] * po[0]) / det
+            print(self.origin, self.axes, point)
+            print(t0, t1)
+            return 0 <= t0 and 0 <= t1 and t0 + t1 <= 1
+        raise NotImplementedError()
 
 
 class Tetrahedron(Reference):
@@ -1203,9 +1293,15 @@ class Tetrahedron(Reference):
         Returns:
             Is the point contained in the reference?
         """
-        if self.vertices != self.reference_vertices:
-            raise NotImplementedError()
-        return 0 <= point[0] and 0 <= point[1] and 0 <= point[2] and sum(point) <= 1
+        if self.vertices == self.reference_vertices:
+            return 0 <= point[0] and 0 <= point[1] and 0 <= point[2] and sum(point) <= 1
+        else:
+            po = _vsub(point, self.origin)
+            minv = sympy.Matrix([[a[i] for a in self.axes] for i in range(3)]).inv()
+            t0 = (minv[0, 0] * po[0] + minv[0, 1] * po[1] + minv[0, 2] * po[2])
+            t1 = (minv[1, 0] * po[0] + minv[1, 1] * po[1] + minv[1, 2] * po[2])
+            t2 = (minv[2, 0] * po[0] + minv[2, 1] * po[1] + minv[2, 2] * po[2])
+            return 0 <= t0 and 0 <= t1 and 0 >= t2 and t0 + t1 + t2 <= 1
 
 
 class Quadrilateral(Reference):


Test patch:
diff --git a/test/test_hct.py b/test/test_hct.py
index fd38eb7..eeadde8 100644
--- a/test/test_hct.py
+++ b/test/test_hct.py
@@ -78,3 +78,20 @@ def test_rhct():
         assert f1.diff(x[0]).diff(x[0]) == 0
         assert f2.diff(x[0]).diff(x[0]) == 0
         assert f3.diff(x[1]).diff(x[1]) == 0
+
+
+def test_rhct_integral():
+    element = symfem.create_element("triangle", "rHCT", 3)
+    ref = element.reference
+    f1 = element.get_basis_function(1).directional_derivative((1, 0))
+    f2 = element.get_basis_function(6).directional_derivative((1, 0))
+    integrand = f1 * f2
+
+    third = sympy.Rational(1, 3)
+    expr = (f1*f2).pieces[((0, 1), (0, 0), (third, third))].as_sympy()
+    assert len((f1*f2).pieces) == 3
+    assert (f1*f2).pieces[((0, 0), (1, 0), (third, third))] == 0
+    assert (f1*f2).pieces[((1, 0), (0, 1), (third, third))] == 0
+
+    assert sympy.integrate(sympy.integrate(
+        expr, (x[1], x[0], 1 - 2 * x[0])), (x[0], 0, third)) == integrand.integral(ref, x)