NURBS Topology#

Modelling a doubly curved NURBS surface bounded by four NURBS lines and reshaping it through its control points:

Create the material and the thickness assigned to the surface
Define the four corner nodes of the surface boundary
Create four NURBS boundary lines, each with three control points whose end points are given by the nodes
Assemble the 3 x 3 control-point matrix of the surface, addressing every row by its u and v position
Create the NURBS surface from the boundary lines and the control-point matrix, with order 3 in both directions
Update a single inner control point by lifting it and increasing its weight to pull the surface towards it

Keywords:

nurbs surface nurbs line control points control point weight surface geometry

import dlubal.api.rfem as rfem
from dlubal.api.common import Vector3d

with rfem.Application() as client:

    client.close_all_models(save_changes=False)
    client.create_model(name="nurbs_surfaces")
    client.delete_all_objects()

    # Material and thickness of the surface.
    client.create_object_list([
        rfem.structure_core.Material(no=1, name="S235"),
        rfem.structure_core.Thickness(no=1, material=1, uniform_thickness=0.020)
    ])

    # Corner nodes of the surface.
    client.create_object_list([
        rfem.structure_core.Node(no=1, global_coordinates=Vector3d(x=11.000, y=0.000, z=0.000)),
        rfem.structure_core.Node(no=2, global_coordinates=Vector3d(x=9.000, y=0.000, z=0.000)),
        rfem.structure_core.Node(no=3, global_coordinates=Vector3d(x=9.000, y=0.000, z=-2.000)),
        rfem.structure_core.Node(no=4, global_coordinates=Vector3d(x=11.000, y=0.000, z=-2.000))
    ])

    # Boundary of a NURBS surface is formed by four NURBS lines.
    # Each of them has three control points, the first and the last one correspond
    # to the end nodes of the line, so we don't need to set any data for them.
    client.create_object_list([
        rfem.structure_core.Line(
            no=1,
            type=rfem.structure_core.Line.TYPE_NURBS,
            definition_nodes=[2, 1],
            nurbs_order=3,
            nurbs_control_points=rfem.structure_core.Line.NurbsControlPointsTable(
                rows=[
                    rfem.structure_core.Line.NurbsControlPointsRow(),
                    rfem.structure_core.Line.NurbsControlPointsRow(global_coordinates=Vector3d(x=10.000, y=0.200, z=0.000), weight=1.000),
                    rfem.structure_core.Line.NurbsControlPointsRow(),
                ]
            )),
        rfem.structure_core.Line(
            no=2,
            type=rfem.structure_core.Line.TYPE_NURBS,
            definition_nodes=[1, 4],
            nurbs_order=3,
            nurbs_control_points=rfem.structure_core.Line.NurbsControlPointsTable(
                rows=[
                    rfem.structure_core.Line.NurbsControlPointsRow(),
                    rfem.structure_core.Line.NurbsControlPointsRow(global_coordinates=Vector3d(x=11.000, y=0.000, z=-1.000), weight=1.000),
                    rfem.structure_core.Line.NurbsControlPointsRow(),
                ]
            )),
        rfem.structure_core.Line(
            no=3,
            type=rfem.structure_core.Line.TYPE_NURBS,
            definition_nodes=[2, 3],
            nurbs_order=3,
            nurbs_control_points=rfem.structure_core.Line.NurbsControlPointsTable(
                rows=[
                    rfem.structure_core.Line.NurbsControlPointsRow(),
                    rfem.structure_core.Line.NurbsControlPointsRow(global_coordinates=Vector3d(x=9.000, y=0.300, z=-1.000), weight=1.000),
                    rfem.structure_core.Line.NurbsControlPointsRow(),
                ]
            )),
        rfem.structure_core.Line(
            no=4,
            type=rfem.structure_core.Line.TYPE_NURBS,
            definition_nodes=[3, 4],
            nurbs_order=3,
            nurbs_control_points=rfem.structure_core.Line.NurbsControlPointsTable(
                rows=[
                    rfem.structure_core.Line.NurbsControlPointsRow(),
                    rfem.structure_core.Line.NurbsControlPointsRow(global_coordinates=Vector3d(x=10.000, y=0.100, z=-2.000), weight=1.000),
                    rfem.structure_core.Line.NurbsControlPointsRow(),
                ]
            ))
    ])

    # Prepare NURBS control points table of the surface.
    # In contrast to the control points of a NURBS line, the control points of a NURBS surface
    # form a 2D matrix, so each row must be addressed by its position in the matrix
    # via the 'u' and 'v' attributes (the 'no' attribute is not used here).
    # The size of the matrix is given by the number of control points in both directions.
    # The rows lying on the boundary of the matrix are defined by the boundary lines,
    # therefore only the inner control point (u=2, v=2) is a new information here.
    # The rows may be listed in any order and rows that are left out keep their current values.
    control_points = rfem.structure_core.Surface.NurbsControlPointsTable(
        rows=[
            rfem.structure_core.Surface.NurbsControlPointsRow(u=1, v=1, global_coordinates=Vector3d(x=11.000, y=0.000, z=-2.000), weight=1.000),
            rfem.structure_core.Surface.NurbsControlPointsRow(u=1, v=2, global_coordinates=Vector3d(x=11.000, y=0.000, z=-1.000), weight=1.000),
            rfem.structure_core.Surface.NurbsControlPointsRow(u=1, v=3, global_coordinates=Vector3d(x=11.000, y=0.000, z=0.000), weight=1.000),
            rfem.structure_core.Surface.NurbsControlPointsRow(u=2, v=1, global_coordinates=Vector3d(x=10.000, y=0.100, z=-2.000), weight=1.000),
            rfem.structure_core.Surface.NurbsControlPointsRow(u=2, v=2, global_coordinates=Vector3d(x=10.000, y=0.100, z=-1.000), weight=1.000),
            rfem.structure_core.Surface.NurbsControlPointsRow(u=2, v=3, global_coordinates=Vector3d(x=10.000, y=0.200, z=0.000), weight=1.000),
            rfem.structure_core.Surface.NurbsControlPointsRow(u=3, v=1, global_coordinates=Vector3d(x=9.000, y=0.000, z=-2.000), weight=1.000),
            rfem.structure_core.Surface.NurbsControlPointsRow(u=3, v=2, global_coordinates=Vector3d(x=9.000, y=0.300, z=-1.000), weight=1.000),
            rfem.structure_core.Surface.NurbsControlPointsRow(u=3, v=3, global_coordinates=Vector3d(x=9.000, y=0.000, z=0.000), weight=1.000),
        ]
    )

    surface = rfem.structure_core.Surface(
        no=1,
        geometry=rfem.structure_core.Surface.GEOMETRY_NURBS,
        boundary_lines=[4, 3, 1, 2],
        thickness=1,
        nurbs_control_point_count_in_direction_u=3,
        nurbs_control_point_count_in_direction_v=3,
        nurbs_order_in_direction_u=3,
        nurbs_order_in_direction_v=3,
        nurbs_control_points=control_points)

    client.create_object(surface)

    # Since every row is addressed by its 'u' and 'v' attributes, a single control point
    # can be modified without repeating the whole matrix.
    # Here we lift the inner control point and increase its weight,
    # which pulls the surface closer to that point.
    client.update_object(
        rfem.structure_core.Surface(
            no=1,
            nurbs_control_points=rfem.structure_core.Surface.NurbsControlPointsTable(
                rows=[
                    rfem.structure_core.Surface.NurbsControlPointsRow(u=2, v=2, global_coordinates=Vector3d(x=10.000, y=0.500, z=-1.000), weight=2.000),
                ]
            )))

using Common = Dlubal.Api.Common;
using Rfem = Dlubal.Api.Rfem;
using Google.Protobuf;

ApplicationRfem? rfemApp = null;

try
{
    rfemApp = new ApplicationRfem();

    rfemApp.close_all_models(saveChanges: false);
    rfemApp.create_model(name: "nurbs_surfaces");
    rfemApp.delete_all_objects();

    // Material and thickness of the surface.
    rfemApp.create_object_list(new List<IMessage>
    {
        new Rfem.StructureCore.Material { No = 1, Name = "S235" },
        new Rfem.StructureCore.Thickness { No = 1, Material = 1, UniformThickness = 0.020 },
    });

    // Corner nodes of the surface.
    rfemApp.create_object_list(new List<IMessage>
    {
        new Rfem.StructureCore.Node { No = 1, GlobalCoordinates = new Common.Vector3d { X = 11.000, Y = 0.000, Z = 0.000 } },
        new Rfem.StructureCore.Node { No = 2, GlobalCoordinates = new Common.Vector3d { X = 9.000, Y = 0.000, Z = 0.000 } },
        new Rfem.StructureCore.Node { No = 3, GlobalCoordinates = new Common.Vector3d { X = 9.000, Y = 0.000, Z = -2.000 } },
        new Rfem.StructureCore.Node { No = 4, GlobalCoordinates = new Common.Vector3d { X = 11.000, Y = 0.000, Z = -2.000 } },
    });

    // Boundary of a NURBS surface is formed by four NURBS lines.
    // Each of them has three control points, the first and the last one correspond
    // to the end nodes of the line, so we don't need to set any data for them.
    rfemApp.create_object_list(new List<IMessage>
    {
        new Rfem.StructureCore.Line
        {
            No = 1,
            Type = Rfem.StructureCore.Line.Types.Type.Nurbs,
            DefinitionNodes = { 2, 1 },
            NurbsOrder = 3,
            NurbsControlPoints = new Rfem.StructureCore.Line.Types.NurbsControlPointsTable
            {
                Rows =
                {
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow(),
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow { GlobalCoordinates = new Common.Vector3d { X = 10.000, Y = 0.200, Z = 0.000 }, Weight = 1.000 },
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow(),
                }
            },
        },
        new Rfem.StructureCore.Line
        {
            No = 2,
            Type = Rfem.StructureCore.Line.Types.Type.Nurbs,
            DefinitionNodes = { 1, 4 },
            NurbsOrder = 3,
            NurbsControlPoints = new Rfem.StructureCore.Line.Types.NurbsControlPointsTable
            {
                Rows =
                {
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow(),
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow { GlobalCoordinates = new Common.Vector3d { X = 11.000, Y = 0.000, Z = -1.000 }, Weight = 1.000 },
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow(),
                }
            },
        },
        new Rfem.StructureCore.Line
        {
            No = 3,
            Type = Rfem.StructureCore.Line.Types.Type.Nurbs,
            DefinitionNodes = { 2, 3 },
            NurbsOrder = 3,
            NurbsControlPoints = new Rfem.StructureCore.Line.Types.NurbsControlPointsTable
            {
                Rows =
                {
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow(),
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow { GlobalCoordinates = new Common.Vector3d { X = 9.000, Y = 0.300, Z = -1.000 }, Weight = 1.000 },
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow(),
                }
            },
        },
        new Rfem.StructureCore.Line
        {
            No = 4,
            Type = Rfem.StructureCore.Line.Types.Type.Nurbs,
            DefinitionNodes = { 3, 4 },
            NurbsOrder = 3,
            NurbsControlPoints = new Rfem.StructureCore.Line.Types.NurbsControlPointsTable
            {
                Rows =
                {
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow(),
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow { GlobalCoordinates = new Common.Vector3d { X = 10.000, Y = 0.100, Z = -2.000 }, Weight = 1.000 },
                    new Rfem.StructureCore.Line.Types.NurbsControlPointsRow(),
                }
            },
        },
    });

    // Prepare NURBS control points table of the surface.
    // In contrast to the control points of a NURBS line, the control points of a NURBS surface
    // form a 2D matrix, so each row must be addressed by its position in the matrix
    // via the 'U' and 'V' attributes (the 'No' attribute is not used here).
    // The size of the matrix is given by the number of control points in both directions.
    // The rows lying on the boundary of the matrix are defined by the boundary lines,
    // therefore only the inner control point (U=2, V=2) is a new information here.
    // The rows may be listed in any order and rows that are left out keep their current values.
    var controlPoints = new Rfem.StructureCore.Surface.Types.NurbsControlPointsTable
    {
        Rows =
        {
            new Rfem.StructureCore.Surface.Types.NurbsControlPointsRow { U = 1, V = 1, GlobalCoordinates = new Common.Vector3d { X = 11.000, Y = 0.000, Z = -2.000 }, Weight = 1.000 },
            new Rfem.StructureCore.Surface.Types.NurbsControlPointsRow { U = 1, V = 2, GlobalCoordinates = new Common.Vector3d { X = 11.000, Y = 0.000, Z = -1.000 }, Weight = 1.000 },
            new Rfem.StructureCore.Surface.Types.NurbsControlPointsRow { U = 1, V = 3, GlobalCoordinates = new Common.Vector3d { X = 11.000, Y = 0.000, Z = 0.000 }, Weight = 1.000 },
            new Rfem.StructureCore.Surface.Types.NurbsControlPointsRow { U = 2, V = 1, GlobalCoordinates = new Common.Vector3d { X = 10.000, Y = 0.100, Z = -2.000 }, Weight = 1.000 },
            new Rfem.StructureCore.Surface.Types.NurbsControlPointsRow { U = 2, V = 2, GlobalCoordinates = new Common.Vector3d { X = 10.000, Y = 0.100, Z = -1.000 }, Weight = 1.000 },
            new Rfem.StructureCore.Surface.Types.NurbsControlPointsRow { U = 2, V = 3, GlobalCoordinates = new Common.Vector3d { X = 10.000, Y = 0.200, Z = 0.000 }, Weight = 1.000 },
            new Rfem.StructureCore.Surface.Types.NurbsControlPointsRow { U = 3, V = 1, GlobalCoordinates = new Common.Vector3d { X = 9.000, Y = 0.000, Z = -2.000 }, Weight = 1.000 },
            new Rfem.StructureCore.Surface.Types.NurbsControlPointsRow { U = 3, V = 2, GlobalCoordinates = new Common.Vector3d { X = 9.000, Y = 0.300, Z = -1.000 }, Weight = 1.000 },
            new Rfem.StructureCore.Surface.Types.NurbsControlPointsRow { U = 3, V = 3, GlobalCoordinates = new Common.Vector3d { X = 9.000, Y = 0.000, Z = 0.000 }, Weight = 1.000 },
        }
    };

    var surface = new Rfem.StructureCore.Surface
    {
        No = 1,
        Geometry = Rfem.StructureCore.Surface.Types.Geometry.Nurbs,
        BoundaryLines = { 4, 3, 1, 2 },
        Thickness = 1,
        NurbsControlPointCountInDirectionU = 3,
        NurbsControlPointCountInDirectionV = 3,
        NurbsOrderInDirectionU = 3,
        NurbsOrderInDirectionV = 3,
        NurbsControlPoints = controlPoints,
    };

    rfemApp.create_object(surface);

    // Since every row is addressed by its 'U' and 'V' attributes, a single control point
    // can be modified without repeating the whole matrix.
    // Here we lift the inner control point and increase its weight,
    // which pulls the surface closer to that point.
    rfemApp.update_object(new Rfem.StructureCore.Surface
    {
        No = 1,
        NurbsControlPoints = new Rfem.StructureCore.Surface.Types.NurbsControlPointsTable
        {
            Rows =
            {
                new Rfem.StructureCore.Surface.Types.NurbsControlPointsRow { U = 2, V = 2, GlobalCoordinates = new Common.Vector3d { X = 10.000, Y = 0.500, Z = -1.000 }, Weight = 2.000 },
            }
        },
    });
}
catch (Exception ex)
{
    Console.WriteLine($"Error: {ex.Message}");
}
finally
{
    if (rfemApp != null) rfemApp.close_connection();
}