<i>This program was contributed by Andrea Mola and Luca Heltai.</i>
-@note This program elaborates on concepts of industrial geometry, using
-tools that interface with the OpenCASCADE library (http://www.opencascade.org) that
-allow the specification of arbitrary IGES files to describe the boundaries for your
-geometries.
+@note This program elaborates on concepts of industrial geometry, using tools
+that interface with the OpenCASCADE library (http://www.opencascade.org) that
+allow the specification of arbitrary IGES files to describe the boundaries for
+your geometries.
<a name="Intro"></a>
<h1>Introduction</h1>
-In some of the previous tutorial programs (step-1, step-3, step-5, step-6 and step-49 among others)
-we have learned how to use the mesh refinement methods provided in deal.II. These tutorials
-have shown how to employ such tools to produce a fine grid for a single simulation, as done
-in step-3; or to start from a coarse grid and carry out a series of simulations on adaptively
-refined grids, as is the case of step-6. Regardless of which approach is taken, the mesh refinement
-requires a suitable geometrical description of the computational domain boundary in order to
-place, at each refinement, the new mesh nodes onto the boundary surface. For instance, step-5
-shows how to assign a circular shape to the boundary of the computational domain, so that
-the faces lying on the boundary are refined onto the circle. step-53 shows how to do this
-with a boundary defined by experimentally obtained data.
-But, at least as far as elementary boundary shapes are concerned, deal.II really only provides
-circles, spheres, boxes and various combinations. In this tutorial, we will show how to use
-a set of classes developed to import
-arbitrary CAD geometries, assign them to the desired boundary of the computational domain,
+In some of the previous tutorial programs (step-1, step-3, step-5, step-6 and
+step-49 among others) we have learned how to use the mesh refinement methods
+provided in deal.II. These tutorials have shown how to employ such tools to
+produce a fine grid for a single simulation, as done in step-3; or to start
+from a coarse grid and carry out a series of simulations on adaptively refined
+grids, as is the case of step-6. Regardless of which approach is taken, the
+mesh refinement requires a suitable geometrical description of the
+computational domain boundary in order to place, at each refinement, the new
+mesh nodes onto the boundary surface. For instance, step-5 shows how to assign
+a circular shape to the boundary of the computational domain, so that the
+faces lying on the boundary are refined onto the circle. step-53 shows how to
+do this with a boundary defined by experimentally obtained data. But, at
+least as far as elementary boundary shapes are concerned, deal.II really only
+provides circles, spheres, boxes and various combinations. In this tutorial,
+we will show how to use a set of classes developed to import arbitrary CAD
+geometries, assign them to the desired boundary of the computational domain,
and refine a computational grid on such complex shapes.
<h3> CAD surfaces </h3>
-In the most common industrial practice, the geometrical models of arbitrarily shaped objects
-are realized by means of Computer Aided Design (CAD) tools. The use of CAD modelers has
-spread in the last decades, as they allow for the generation of a full virtual model of
-each designed object, which through a computer can be visualized, inspected, and analyzed
-in its finest details well before it is physically crafted.
-From a mathematical perspective, the engine lying under the hood of CAD modelers is represented
-by analytical geometry, and in particular by parametric curves and surfaces such as B-splines
-and NURBS that are rich enough that they can represent most surfaces of practical interest.
-Once a virtual model is ready, all the geometrical features of the desired object
-are stored in files which materially contain the coefficients of the parametric surfaces and
-curves composing the object. Depending on the specific CAD tool used to define the geometrical model,
-there are of course several different file formats in which the information of a CAD model can
-be organized. To provide a common ground to exchange data across CAD tools, the U.S. National
-Bureau of Standards published in 1980 the Initial Graphics Exchange Representation (IGES) neutral
-file format, which is used in this example.
+In the most common industrial practice, the geometrical models of arbitrarily
+shaped objects are realized by means of Computer Aided Design (CAD) tools. The
+use of CAD modelers has spread in the last decades, as they allow for the
+generation of a full virtual model of each designed object, which through a
+computer can be visualized, inspected, and analyzed in its finest details well
+before it is physically crafted. From a mathematical perspective, the engine
+lying under the hood of CAD modelers is represented by analytical geometry,
+and in particular by parametric curves and surfaces such as B-splines and
+NURBS that are rich enough that they can represent most surfaces of practical
+interest. Once a virtual model is ready, all the geometrical features of the
+desired object are stored in files which materially contain the coefficients
+of the parametric surfaces and curves composing the object. Depending on the
+specific CAD tool used to define the geometrical model, there are of course
+several different file formats in which the information of a CAD model can be
+organized. To provide a common ground to exchange data across CAD tools, the
+U.S. National Bureau of Standards published in 1980 the Initial Graphics
+Exchange Representation (IGES) neutral file format, which is used in this
+example.
<h3> The boundary projector classes </h3>
To import and interrogate CAD models, the deal.II library contains a series of
-wrapper functions for the OpenCASCADE open source library for CAD modeling. These functions allow to import
-IGES files into OpenCASCADE native objects, and wrap them inside a series of Manifold classes.
-
-Once imported from an IGES file, the model is stored in a <code>TopoDS_Shape</code>, which is the generic topological entity
-defined in the OpenCASCADE framework. From a <code>TopoDS_Shape</code>, it is then possible to access all the
-sub-shapes (such as vertices, edges and faces) composing it, along with their geometrical description.
-In the deal.II framework, the topological entities composing a shape are used to create objects
-of the Manifold or Boundary classes. In Step-6 we saw how to build a HyperBallBoundary and assign
-it to a set of faces (or cells, for co-dimension 1) of a Triangulation, to have cells and faces
-refined on a sphere or circle.
-The functions of the CAD modeling interface have been designed to retain the same structure, allowing the user to build a
-projector object using the imported CAD shapes, maintaining the
-very same procedure we use with HyperBallBoundary, i.e., assigning such projector object to cells,
-faces or edges of a coarse mesh. At each refinement cycle, the new mesh nodes will be then automatically generated
-by projecting a midpoint of an existing object onto the specified geometry.
-
-Differently from a spherical or circular boundary, a boundary with a complex geometry
-poses proplems as to where it is best to place the new nodes created upon refinement on the prescribed shape.
-HyperBallBoundary first creates the new nodes on the face or edge to be refined by averaging the
-surrounding points in the same way as FlatManifold does. Then, it goes on to project such nodes on the circle or
-sphere along the radial direction. On such a geometry, the radial direction ensures that the newly generated nodes remain evenly
-spaced when remaining on a given refinement level.
-
-In the case of an arbitrary and complex shape though, the best direction of the projection cannot be identified that easily.
-The OpenCASCADE wrappers in deal.II provide several projector classes that employ different projection strategies. A first
-projector, implemented in the OpenCASCADE::ArclengthProjectionLineManifold class, is to be used only for edge refinement. It is built assigning it a topological shape
-of dimension one, either a <code>TopoDS_Edge</code> or a <code>TopoDS_Wire</code> (which is a compound shape, made of several connected <code>TopoDS_Edge</code>s)
-and refines a mesh edge finding the new vertex as the point splitting in two even parts the curvilinear length of the CAD curve
-portion that lies between the vertices of the original edge.
+wrapper functions for the OpenCASCADE open source library for CAD
+modeling. These functions allow to import IGES files into OpenCASCADE native
+objects, and wrap them inside a series of Manifold classes.
+
+Once imported from an IGES file, the model is stored in a
+<code>TopoDS_Shape</code>, which is the generic topological entity defined in
+the OpenCASCADE framework. From a <code>TopoDS_Shape</code>, it is then
+possible to access all the sub-shapes (such as vertices, edges and faces)
+composing it, along with their geometrical description. In the deal.II
+framework, the topological entities composing a shape are used to create
+objects of the Manifold or Boundary classes. In Step-6 we saw how to build a
+HyperBallBoundary and assign it to a set of faces (or cells, for co-dimension
+1) of a Triangulation, to have cells and faces refined on a sphere or circle.
+The functions of the CAD modeling interface have been designed to retain the
+same structure, allowing the user to build a projector object using the
+imported CAD shapes, maintaining the very same procedure we use with
+HyperBallBoundary, i.e., assigning such projector object to cells, faces or
+edges of a coarse mesh. At each refinement cycle, the new mesh nodes will be
+then automatically generated by projecting a midpoint of an existing object
+onto the specified geometry.
+
+Differently from a spherical or circular boundary, a boundary with a complex
+geometry poses proplems as to where it is best to place the new nodes created
+upon refinement on the prescribed shape. HyperBallBoundary first creates the
+new nodes on the face or edge to be refined by averaging the surrounding
+points in the same way as FlatManifold does. Then, it goes on to project such
+nodes on the circle or sphere along the radial direction. On such a geometry,
+the radial direction ensures that the newly generated nodes remain evenly
+spaced when remaining on a given refinement level.
+
+In the case of an arbitrary and complex shape though, the best direction of
+the projection cannot be identified that easily. The OpenCASCADE wrappers in
+deal.II provide several projector classes that employ different projection
+strategies. A first projector, implemented in the
+OpenCASCADE::ArclengthProjectionLineManifold class, is to be used only for
+edge refinement. It is built assigning it a topological shape of dimension
+one, either a <code>TopoDS_Edge</code> or a <code>TopoDS_Wire</code> (which is
+a compound shape, made of several connected <code>TopoDS_Edge</code>s) and
+refines a mesh edge finding the new vertex as the point splitting in two even
+parts the curvilinear length of the CAD curve portion that lies between the
+vertices of the original edge.
<img src="http://www.dealii.org/images/steps/developer/step-54.CurveSplit.png" alt="" width="500">
-A different projection stategy has been implemented in the OpenCASCADE::NormalProjectionBoundary class. The <code>TopoDS_Shape</code> assigned at construction time can be arbitrary
-(a collection of shapes, faces, edges or a single face or edge will all work). The new cell nodes are first computed by averaging the
-surrounding points in the same way as FlatManifold does. In a second step, all the new nodes will be projected onto the <code>TopoDS_Shape</code>
-along the direction normal to the shape. If no normal projection is available, the point which is closest to the shape---typically lying on the shape boundary---is selected.
-If the shape is composed of several sub-shapes, the projection is carried out onto every single
-sub-shape, and the closest projection point point is selected.
+A different projection stategy has been implemented in the
+OpenCASCADE::NormalProjectionBoundary class. The <code>TopoDS_Shape</code>
+assigned at construction time can be arbitrary (a collection of shapes, faces,
+edges or a single face or edge will all work). The new cell nodes are first
+computed by averaging the surrounding points in the same way as FlatManifold
+does. In a second step, all the new nodes will be projected onto the
+<code>TopoDS_Shape</code> along the direction normal to the shape. If no
+normal projection is available, the point which is closest to the
+shape---typically lying on the shape boundary---is selected. If the shape is
+composed of several sub-shapes, the projection is carried out onto every
+single sub-shape, and the closest projection point point is selected.
<img src="http://www.dealii.org/images/steps/developer/step-54.NormalProjectionEdge.png" alt="" width="500">
<img src="http://www.dealii.org/images/steps/developer/step-54.NormalProjection.png" alt="" width="500">
-As we are about to experience, for some shapes, setting the projection direction as that normal to the CAD surface will not
-lead to surface mesh elements of suitable quality. This is because the direction normal to the CAD surface has in principle nothing to do with the
-direction along which the mesh needs the new nodes to be located. The OpenCASCADE::DirectionalProjectionBoundary class, in this case, can help. This class
-is constructed assigning a <code>TopoDS_Shape</code> (containing at least a face) and a direction along which all the projections will be carried out. New points will be
-computed by first averaging the surrounding points (as in the FlatManifold case), and then taking the closest intersection between the topological shape
-and the line passing throught the resulting point, along the direction used at construction time.
-In this way, the user will have a higher control on the projection direction to be enforced to ensure good mesh quality.
+As we are about to experience, for some shapes, setting the projection
+direction as that normal to the CAD surface will not lead to surface mesh
+elements of suitable quality. This is because the direction normal to the CAD
+surface has in principle nothing to do with the direction along which the mesh
+needs the new nodes to be located. The
+OpenCASCADE::DirectionalProjectionBoundary class, in this case, can help. This
+class is constructed assigning a <code>TopoDS_Shape</code> (containing at
+least a face) and a direction along which all the projections will be carried
+out. New points will be computed by first averaging the surrounding points (as
+in the FlatManifold case), and then taking the closest intersection between
+the topological shape and the line passing throught the resulting point, along
+the direction used at construction time. In this way, the user will have a
+higher control on the projection direction to be enforced to ensure good mesh
+quality.
<img src="http://www.dealii.org/images/steps/developer/step-54.DirectionalProjection.png" alt="" width="500">
-Of course the latter approach is effective only when the orientation of the surface is rather uniform, so that a single projection
-direction can be identified. In cases in which the surface direction is approaching the projection direction, it is even possible that the
-directional projection is not found. To overcome these problems, the OpenCASCADE::NormalToMeshProjectionBoundary class implements a third
-projection algorithm. The OpenCASCADE::NormalToMeshProjectionBoundary class is built assigning a <code>TopoDS_Shape</code> (containing at least one face) to the contructor,
-and works exactly like a OpenCASCADE::DirectionalProjection. But, as the name of the class suggests, OpenCASCADE::NormalToMeshProjectionBoundary tries to come up
-with a suitable estimate of the direction normal to the mesh elements to be refined, and uses it for the projection of the new nodes
-onto the CAD surface. If we consider a mesh edge in a 2D space, the direction of its axis is a direction along which to split it in order to
-give rise to two new cells of the same length. We here extended this concept in 3D, and project all new nodes in a direction that approximates
-the cell normal.
-
-In the next figure, which is inpired by the geometry considered in this tutorial, we make an attempt to compare the behavior of the
-three projectors considered. As can be seen on the left, given the original cell (in blue), the new point found with the normal projection is
-in a position which does not allow for the generation of evenly spaced new elements (in red). The situation will get worse in further refinement steps.
-Since the geometry we considered is somehow perpendicular to the horizontal direction, the directional projection (central image) defined
-with horizontal direction as the projection direction, does a rather good job in getting the new mesh point. Yet, since the surface is almost horizontal
-at the bottom of the picture, we can expect problems in those regions when further refinement steps are carried out. Finally,
-the picture on the right shows that a node located on the cell axis will result in two new cells having the same length. Of course the situation
-in 3D gets a little more complicated than that described in this simple 2D case. Nevertheless, the results of this test confirm that the normal to the
-mesh direction is the best approach among the three tested, when arbitrarily shaped surfaces are considered, and unless you have a geometry for which a more specific approach is known to be appropriate.
+Of course the latter approach is effective only when the orientation of the
+surface is rather uniform, so that a single projection direction can be
+identified. In cases in which the surface direction is approaching the
+projection direction, it is even possible that the directional projection is
+not found. To overcome these problems, the
+OpenCASCADE::NormalToMeshProjectionBoundary class implements a third
+projection algorithm. The OpenCASCADE::NormalToMeshProjectionBoundary class is
+built assigning a <code>TopoDS_Shape</code> (containing at least one face) to
+the contructor, and works exactly like a
+OpenCASCADE::DirectionalProjection. But, as the name of the class suggests,
+OpenCASCADE::NormalToMeshProjectionBoundary tries to come up with a suitable
+estimate of the direction normal to the mesh elements to be refined, and uses
+it for the projection of the new nodes onto the CAD surface. If we consider a
+mesh edge in a 2D space, the direction of its axis is a direction along which
+to split it in order to give rise to two new cells of the same length. We here
+extended this concept in 3D, and project all new nodes in a direction that
+approximates the cell normal.
+
+In the next figure, which is inpired by the geometry considered in this
+tutorial, we make an attempt to compare the behavior of the three projectors
+considered. As can be seen on the left, given the original cell (in blue), the
+new point found with the normal projection is in a position which does not
+allow for the generation of evenly spaced new elements (in red). The situation
+will get worse in further refinement steps. Since the geometry we considered
+is somehow perpendicular to the horizontal direction, the directional
+projection (central image) defined with horizontal direction as the projection
+direction, does a rather good job in getting the new mesh point. Yet, since
+the surface is almost horizontal at the bottom of the picture, we can expect
+problems in those regions when further refinement steps are carried
+out. Finally, the picture on the right shows that a node located on the cell
+axis will result in two new cells having the same length. Of course the
+situation in 3D gets a little more complicated than that described in this
+simple 2D case. Nevertheless, the results of this test confirm that the normal
+to the mesh direction is the best approach among the three tested, when
+arbitrarily shaped surfaces are considered, and unless you have a geometry for
+which a more specific approach is known to be appropriate.
<img src="http://www.dealii.org/images/steps/developer/step-54.ProjectionComparisons.png" alt="" width="700">
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