Create a domain for an Hexahedron with the floor replaced with an Indian Burial Mound

Question

I am trying to create an acoustic model domain for a small park that contains an Indian burial mound.

• The park is laid out using a 3D cartesian coordinate system in units of meters.
• A Hexahedron encapsulates the park. The ceiling of the Hexahedron is 16 meters (about the height of a 5-story building).
• The {x, y, z} coordinates of the burial mound are in meters. The burial mound coordinates are imported from am Excel workbook.

My initial approach is to define the hexahedron that represents the volumetric domain that I want to simulate around the burial mound.

I defined the hexahedron as:

hexDomain = Hexahedron[{{0,0,0},  {40,0,0}, {40,35,0},{0,35,0},
                        {0,0,16}, {40,0,16},{40,35,16},{0,35,16}}];

I then import the points with the coordinates from the Excel file with:

moundPoints = Import["C:\\Spreadsheets\\Mound Coordinates.xlsx",
                     {"Data", 1, 2;;82, {5,6,7}}];

ListSurfacePlot3D[moundPoints] of the mound is below:

My goal is to replace floor of the volumetric hexDomain with the burial mound as a solid.

I started by creating a solid region for the burial mound with:

moundRegion = ConvexHullRegion[moundPoints];

As you might expect, a RegionDifference between hexDomain and moundRegion will introduce lines and slices between the convex hull points and the hexahedron.

My question is:

What is the best approach to create the domain that includes the burial mound inside of a five-story hexahedron?

If I add the faces of the hexahedron to the moundPoints, I get a domain with connecting lines that should not exist.

The data from Mound Coordinates.xlsx is below.

moundPoints = {{14.0667, 32.719, 0.}, {12.8476, 32.1037, 0.}, {8.9873, 29.9502, 
  0.}, {5.12698, 28.0018, 0.}, {5.22857, 27.7967, 0.}, {1.57143, 
  25.6431, 0.}, {0.75873, 24.8227, 0.}, {1.87619, 17.9519, 
  0.}, {2.18095, 12.7219, 0.}, {5.0254, 9.54292, 0.}, {8.07302, 
  6.26135, 0.}, {11.7302, 3.28743, 0.}, {16.6063, 5.03076, 
  0.}, {19.5524, 7.90214, 0.}, {23.3111, 10.2608, 0.}, {25.0381, 
  11.0812, 0.}, {23.4127, 10.0557, 0.}, {26.2571, 13.85, 
  0.}, {27.4762, 18.9774, 0.}, {24.327, 23.5921, 0.}, {22.1937, 
  27.0788, 0.}, {18.1302, 29.9502, 0.}, {15.4889, 28.3094, 
  1.}, {11.3238, 27.0788, 1.}, {7.97143, 24.0023, 1.}, {6.65079, 
  18.5672, 1.}, {7.05714, 16.5163, 1.}, {10.1048, 13.0296, 
  1.}, {14.5746, 11.2863, 1.}, {17.6222, 10.7735, 1.}, {21.5841, 
  12.7219, 1.}, {24.0222, 16.6188, 1.}, {24.6317, 18.2596, 
  1.}, {23.6159, 23.0794, 1.}, {20.9746, 26.2584, 1.}, {15.7937, 
  28.0018, 1.}, {13.8635, 26.8737, 2.}, {12.0349, 26.0533, 
  2.}, {9.19048, 22.4641, 2.}, {8.37778, 20.4131, 2.}, {8.47937, 
  17.5418, 2.}, {11.1206, 13.85, 2.}, {13.254, 12.927, 2.}, {16.6063, 
  12.1067, 2.}, {18.7397, 12.2092, 2.}, {20.9746, 13.6449, 
  2.}, {22.9048, 18.157, 2.}, {22.7016, 22.0539, 2.}, {21.4825, 
  24.1049, 2.}, {19.4508, 25.3355, 2.}, {16.5048, 26.2584, 
  2.}, {18.3333, 24.5151, 3.}, {13.9651, 25.6431, 3.}, {12.746, 
  25.0278, 3.}, {11.6286, 24.31, 3.}, {10.1048, 21.3361, 3.}, {9.8, 
  18.3621, 3.}, {12.2381, 14.7729, 3.}, {15.5905, 13.6449, 
  3.}, {19.0444, 13.6449, 3.}, {20.2635, 14.5678, 3.}, {21.4825, 
  17.4392, 3.}, {21.6857, 20.0029, 3.}, {21.4825, 21.2335, 
  3.}, {20.4667, 23.0794, 3.}, {18.2317, 24.31, 3.}, {17.0127, 
  23.4896, 4.}, {15.1841, 24.2074, 4.}, {13.4571, 24.0023, 
  4.}, {11.2222, 19.9004, 4.}, {11.3238, 18.6698, 4.}, {13.9651, 
  15.7984, 4.}, {18.3333, 14.8755, 4.}, {19.2476, 15.901, 
  4.}, {20.0603, 20.1055, 4.}, {18.9429, 22.259, 4.}, {17.1143, 
  18.5672, 4.8}, {16.8095, 17.8494, 4.7}, {16.9111, 17.2341, 
  4.8}, {17.5206, 18.5672, 4.8}, {17.5206, 19.1825, 4.7}}

Answer 1

There are a few very minor issue in Doug Kimzey's answer.

1. The mound is concave, not convex.

2. It did not complete the journey to an ElementMesh.

As I said, these are minor issues.

Onward, using an extremely similar example in Wolfram Mathematica's Element Mesh Generation tutorial, here is an example using the mound data.

Timing[moundPoints = {{14.0667, 32.719, 0.}, {12.8476, 32.1037, 0.}, 
     {8.9873, 29.9502, 0.}, {5.12698, 28.0018, 0.}, 
     {5.22857, 27.7967, 0.}, {1.57143, 25.6431, 0.}, 
     {0.75873, 24.8227, 0.}, {1.87619, 17.9519, 0.}, 
     {2.18095, 12.7219, 0.}, {5.0254, 9.54292, 0.}, 
     {8.07302, 6.26135, 0.}, {11.7302, 3.28743, 0.}, 
     {16.6063, 5.03076, 0.}, {19.5524, 7.90214, 0.}, 
     {23.3111, 10.2608, 0.}, {25.0381, 11.0812, 0.}, 
     {23.4127, 10.0557, 0.}, {26.2571, 13.85, 0.}, 
     {27.4762, 18.9774, 0.}, {24.327, 23.5921, 0.}, 
     {22.1937, 27.0788, 0.}, {18.1302, 29.9502, 0.}, 
     {15.4889, 28.3094, 1.}, {11.3238, 27.0788, 1.}, 
     {7.97143, 24.0023, 1.}, {6.65079, 18.5672, 1.}, 
     {7.05714, 16.5163, 1.}, {10.1048, 13.0296, 1.}, 
     {14.5746, 11.2863, 1.}, {17.6222, 10.7735, 1.}, 
     {21.5841, 12.7219, 1.}, {24.0222, 16.6188, 1.}, 
     {24.6317, 18.2596, 1.}, {23.6159, 23.0794, 1.}, 
     {20.9746, 26.2584, 1.}, {15.7937, 28.0018, 1.}, 
     {13.8635, 26.8737, 2.}, {12.0349, 26.0533, 2.}, 
     {9.19048, 22.4641, 2.}, {8.37778, 20.4131, 2.}, 
     {8.47937, 17.5418, 2.}, {11.1206, 13.85, 2.}, 
     {13.254, 12.927, 2.}, {16.6063, 12.1067, 2.}, 
     {18.7397, 12.2092, 2.}, {20.9746, 13.6449, 2.}, 
     {22.9048, 18.157, 2.}, {22.7016, 22.0539, 2.}, 
     {21.4825, 24.1049, 2.}, {19.4508, 25.3355, 2.}, 
     {16.5048, 26.2584, 2.}, {18.3333, 24.5151, 3.}, 
     {13.9651, 25.6431, 3.}, {12.746, 25.0278, 3.}, 
     {11.6286, 24.31, 3.}, {10.1048, 21.3361, 3.}, {9.8, 18.3621, 3.}, 
     {12.2381, 14.7729, 3.}, {15.5905, 13.6449, 3.}, 
     {19.0444, 13.6449, 3.}, {20.2635, 14.5678, 3.}, 
     {21.4825, 17.4392, 3.}, {21.6857, 20.0029, 3.}, 
     {21.4825, 21.2335, 3.}, {20.4667, 23.0794, 3.}, 
     {18.2317, 24.31, 3.}, {17.0127, 23.4896, 4.}, 
     {15.1841, 24.2074, 4.}, {13.4571, 24.0023, 4.}, 
     {11.2222, 19.9004, 4.}, {11.3238, 18.6698, 4.}, 
     {13.9651, 15.7984, 4.}, {18.3333, 14.8755, 4.}, 
     {19.2476, 15.901, 4.}, {20.0603, 20.1055, 4.}, 
     {18.9429, 22.259, 4.}, {17.1143, 18.5672, 4.8}, 
     {16.8095, 17.8494, 4.7}, {16.9111, 17.2341, 4.8}, 
     {17.5206, 18.5672, 4.8}, {17.5206, 19.1825, 4.7}}; 
   mound = DiscretizeGraphics[ListPlot3D[Join[moundPoints, 
       {{0, 2, 0}, {28, 2, 0}, {0, 34, 0}, {28, 34, 0}}], 
      Boxed -> False, Axes -> False, Mesh -> None, PlotRange -> All]]; 
   building = Select[MeshPrimitives[Cuboid[{0, 2, 0}, {28, 34, 7.2}], 
      2], #1 =!= Polygon[N[{{0, 34, 0}, {28, 34, 0}, {28, 2, 0}, 
          {0, 2, 0}}]] & ];
region = RegionUnion[Append[building, mound]]]

This took 15.625 milliseconds on my machine.

Yes, it makes an ElementMesh ( ToElementMesh[region]["Wireframe"] ).

The machine:

13.2.1 for Microsoft Windows (64-bit) (January 27, 2023) Microsoft Windows 10 Pro 10.0.19045
Intel(R) Core(TM) i7-9700K CPU @ 3.60GHz
{NumberOfCores,8} {DisplayVersion,22H2}

Of course, I may have missed something.

Answer 2

Does this work for you?

parkReg = RegionDifference[Region[hexDomain], ConvexHullRegion[moundPoints]]
ConstantRegionQ[parkReg]
Volume[parkReg]

(* True *)
(* 21335. *)