Reference: Deformable Surface

Model Element

Description

Reference_DeformSurface element specifies a deformable surface that is made to pass through the origins of a specified set of markers, using CUBIC spline interpolation. These markers may be on separate bodies or on a flexible body. As the markers move in space, the surface is recalculated using CUBIC spline interpolation thereby allowing the surface to deform. The surface can be open or closed.

Format

<Reference_DeformSurface

id = “integer”

[ label = “string” ]

u_span = “real”

v_span = “real”

[ end_type = “{NATURAL | PARABOLIC | PERIODIC | CANTILEVER}” ]

[ is_u_closed = “boolean” ]

[ is_v_closed = “boolean” ]

num_marker_row = “integer”

num_marker_col = “integer”>

integer integer … integer

… … … …

integer integer … integer

</Reference_DeformSurface>

Attributes

Specifies the element identification number, (integer>0). This number should be unique among all Reference_DeformSurface elements.

label

An optional character string that defines a name for the modeling element.

u_span

Real value that defines the span of the u parameter. The u parameter goes from –u_span/2 to u_span/2. u_span > 0.

v_span

Real value that defines the span of the v parameter. The v parameter goes from –v_span/2 to v_span/2. v_span > 0.

end_type

Select one from NATURAL, PARABOLIC or PERIODIC. See Comment 8 for more details.

Default = NATURAL

is_u_closed

A Boolean value (TRUE or FALSE) that determines whether the markers that define the surface are closed in u or not. See Comments 3, 4 for more details.

Default = “FALSE”

is_v_closed

A Boolean value (TRUE or FALSE) that determines whether the markers that define the surface are closed in v or not. See Comments 3, 5 for more details.

Default = “FALSE”

num_marker_row

Number of markers that define the u curves. See Comment 7 for more details.

num_marker_col

Number of markers that define the v curves. See Comment 7 for more details.

Comments

1.	Reference_DeformSurface is quite similar to Reference_ParamSurface, except it can deform.

refdeform_model_img1

Figure 1: The definition of a Reference_DeformSurface.

Referring to Figure 1, the shape of the surface S is defined by the cubic spline that passes through the instantaneous origins of a set of markers. The origins are shown as green spheres in Figure 1. Reference_DeformSurface is therefore a parametric, deformable surface in 3-D space. u and v are free surface parameters that define the extent of the surface.

•	u can vary from umin = - u_span/2 to umax = + u_span/2

•	v can vary from vmin = - v_span/2 to vmax = + v_span/2

The coordinates of any arbitrary P point on the surface, as measured in OXYZ, can be represented uniquely in terms of the free parameters u, v, with three cubic-spline functions f(u,v), g(u,v) and h(u,v) that define the x-, y- and z-coordinates of P.

2.	The markers used to define the u and v curves are arranged as follows:

Marker matrix:

refdeform_model_img2

such that the total number of markers for (u x v) = (n x m).

3.	Reference_DeformCurve can be opened or closed in either the u or v domains.

•	A surface closed in u has the property:

o	{ x(umin,v) y(umin,v) z(umin,v) } = { x(umax,v) y(umax,v) z(umax,v) }

•	A surface closed in v has the property:

o	{ x(u,vmin) y(u,vmin) z(u,vmin) } = { x(u,vmax) y(u,vmax) z(u,vmax) }

•	A surface closed in u and v has the properties:

o	{ x(umin,v) y(umin,v) z(umin,v) } = { x(umax,v) y(umax,v) z(umax,v) }, and,

o	{ x(u,vmin) y(u,vmin) z(u,vmin) } = { x(u,vmax) y(u,vmax) z(u,vmax) }

4.	To define the surface as closed in the u space, you must do the following:

•	Set is_u_closed = “TRUE”

•	Ensure that the first and last markers in every column are the same.

•	See Example 2a for reference.

5.	To define the surface as closed in the v space, you must do the following:

•	Set is_v_closed = “TRUE”

•	Ensure that the first and last markers in every row are the same.

•	See Example 2b for reference.

6.	To define the surface as closed in both the u and v space, you must do the following:

•	Set is_u_closed = “TRUE”

•	Set is_v_closed = “TRUE”

•	Ensure that the first and last markers in every column are the same.

•	Ensure that the first and last markers in every row are the same.

•	See Example 2c for reference.

7.	The minimum number of markers required to define a surface is listed below:

refdeform_model_img3

8.	Reference_DeformSurface uses a CUBIC spline interpolation that requires assumptions on the second derivative of the interpolating function at the ends of the surface. The keywords NATURAL, PARABOLIC, and PERIODIC represent the three standard assumptions defined as follows:

NATURAL (or free):

PARABOLIC:

PERIODIC:

9.	The Reference_DeformSurface element does not possess any inherent inertia, stiffness or damping properties.

10.	A deformable surface is used in the definition of a Constraint_PTdSF, which constrains a point on a body to slide along the deformable surface.

11.	This deformable surface is also used in the definition of a Force_PTdSF, which prevents a point on a body from penetrating into the deformable surface, but allows lift-off and sliding.

12.

If the Reference_DeformSurface is used for a Force_PTdSF element, then the surface normal is required. The surface normal is calculated at a point on the surface as the cross product between the tangent to the u curve and the tangent to the v curve at that point. The attribute flip_normal can be used to flip the direction of the normal vector. See the Force_PTdSF element for more details.

13.

Occasionally, for an open deformable surface (open in U, open in V or open in both U and V), you will see the message,
"Parameter U=’value’ goes out of intended range". This condition occurs because the numerical methods have no obvious way to enforce the limit: – u_span/2 ≤ u ≤ + u_span/2 and – v_span/2 ≤ v ≤ + v_span/2. In such a situation, MotionSolve holds the "U" or "V" value at the boundary (-u_span/2 or +u_span/2 for u and -v_span/2 or +v_span/2 for v) and continues with the simulation until the point falls back in range. If the model is well-defined, the point will fall back in range.

Note: If you see such a message in your simulation, please check the results carefully as they may be wrong, even though the simulation completes successfully.

Example 1

Figure 2 below demonstrates the use of the Constraint_PTDSF element, along with the Reference_DeformSurface and Post_Graphic elements.

refdeform_model_img4

Figure 2: A body constrained to move on a deformable surface.

<Reference_DeformSurface

id = "1"

end_type = "NATURAL"

num_marker_row = "7"

num_marker_col = "7">

30101490 30101500 30101510 30101520 30101530 30101540 30101540

30101420 30109781 30113791 30117801 30121811 30125821 30101480

30101350 30108711 30112721 30116731 30120741 30124751 30101410

30101120 30102061 30103071 30104081 30105091 30106101 30101130

30101140 30110571 30114581 30118591 30122601 30126611 30101200

30101210 30111641 30115651 30119661 30123671 30127681 30101270

30101280 30101290 30101300 30101310 30101320 30101330 30101340

</Reference_DeformSurface>

<Constraint_PTdSF

id = "1"

i_marker_id = "30107780"

ref_surface_id = "1"

<Post_Graphic

id = "500000"

type = "DeformSurface"

surface_id = "1"

nseg_u = "30"

nseg_v = "30"

Example 2

Figures 3-5 below, demonstrate how to define a closed surface. In the images, the direction of U, V is shown and the deformable surface is labeled as S.

a)	Closed in u:

refdeform_model_img5

Figure 3: A deformable surface closed in u space, but not in v.

<Reference_DeformSurface

id = "301001"

label = "DeformableSurface 0"

end_type = "NATURAL"

u_span = "1."

v_span = "1."

is_u_closed = "TRUE"

is_v_closed = "FALSE"

num_marker_row = "13"

num_marker_col = "12">

70000000 30102970 30102740 30102500 30102260 30102140 30102020 30102380 70000002 30102620 30102860 70000003

70000005 30102980 30102750 30102510 30102270 30102150 30102030 30102390 70000006 30102630 30102870 70000007

70000009 30102990 30102760 30102520 30102280 30102160 30102040 30102400 70000010 30102640 30102880 70000011

70000013 70000014 30102770 30102530 30102290 30102170 30102050 30102410 70000015 30102650 30102890 70000016

70000018 70000019 30102780 30102540 30102300 30102180 30102060 30102420 70000020 30102660 30102900 70000021

70000023 70000024 30102790 30102550 30102310 30102190 30102070 30102430 70000025 30102670 30102910 70000026

70000028 70000029 30102800 30102560 30102320 30102200 30102080 30102440 70000030 30102680 30102920 70000031

70000033 70000034 30102810 30102570 30102330 30102210 30102090 30102450 70000035 30102690 70000036 70000037

70000039 70000040 30102820 30102580 30102340 30102220 30102110 30102460 70000041 30102700 30102930 70000042

70000044 70000045 30102830 30102590 30102350 30102230 30102100 30102470 70000046 30102710 30102940 70000047

70000049 70000050 30102840 30102600 30102360 30102240 30102120 30102480 70000051 30102720 30102950 70000052

70000054 70000055 30102850 30102610 30102370 30102250 30102130 30102490 70000056 30102730 30102960 70000057

70000000 30102970 30102740 30102500 30102260 30102140 30102020 30102380 70000002 30102620 30102860 70000003

</Reference_DeformSurface>

b)	Closed in v:

refdeform_model_img6

Figure 4: A deformable surface closed in v space, but not in u.

<Reference_DeformSurface

id = "301001"

label = "DeformableSurface 0"

end_type = "NATURAL"

u_span = "1."

v_span = "1."

is_u_closed = "FALSE"

is_v_closed = "TRUE"

num_marker_row = "12"

num_marker_col = "13">

70000000 30102970 30102740 30102500 30102260 30102140 30102020 30102380 70000002 30102620 30102860 70000003 70000000

70000005 30102980 30102750 30102510 30102270 30102150 30102030 30102390 70000006 30102630 30102870 70000007 70000005

70000009 30102990 30102760 30102520 30102280 30102160 30102040 30102400 70000010 30102640 30102880 70000011 70000009

70000013 70000014 30102770 30102530 30102290 30102170 30102050 30102410 70000015 30102650 30102890 70000016 70000013

70000018 70000019 30102780 30102540 30102300 30102180 30102060 30102420 70000020 30102660 30102900 70000021 70000018

70000023 70000024 30102790 30102550 30102310 30102190 30102070 30102430 70000025 30102670 30102910 70000026 70000023

70000028 70000029 30102800 30102560 30102320 30102200 30102080 30102440 70000030 30102680 30102920 70000031 70000028

70000033 70000034 30102810 30102570 30102330 30102210 30102090 30102450 70000035 30102690 70000036 70000037 70000033

70000039 70000040 30102820 30102580 30102340 30102220 30102110 30102460 70000041 30102700 30102930 70000042 70000039

70000044 70000045 30102830 30102590 30102350 30102230 30102100 30102470 70000046 30102710 30102940 70000047 70000044

70000049 70000050 30102840 30102600 30102360 30102240 30102120 30102480 70000051 30102720 30102950 70000052 70000049

70000054 70000055 30102850 30102610 30102370 30102250 30102130 30102490 70000056 30102730 30102960 70000057 70000054

</Reference_DeformSurface>

c)	Closed in both u and v:

refdeform_model_img7

Figure 5: A deformable surface closed in u and v space.