Skip to content

Sdf

ontic.cfd.sdf

Signed Distance Function (SDF) geometry library.

Provides a protocol-based geometry abstraction for immersed boundary methods on structured Cartesian grids. Each SDF maps every point in space to its signed distance from the nearest surface:

d < 0  →  inside solid
d = 0  →  on the surface
d > 0  →  outside (fluid)

All implementations are torch-compatible and differentiable for gradient-based shape optimization.

References
  • Mittal & Iaccarino, "Immersed Boundary Methods", Annu. Rev. Fluid Mech. 37:239-261, 2005
  • Hart, "Sphere Tracing", Visual Comp. 12(10), 1996

SDFGeometry

Bases: ABC

Abstract base for 2-D signed-distance-function geometries.

Every concrete subclass must implement sdf. The default is_inside, normal, and surface_points derive from sdf automatically but may be overridden for efficiency.

Source code in ontic/cfd/sdf.py 
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
class SDFGeometry(abc.ABC):
    """Abstract base for 2-D signed-distance-function geometries.

    Every concrete subclass must implement ``sdf``.  The default
    ``is_inside``, ``normal``, and ``surface_points`` derive from
    ``sdf`` automatically but may be overridden for efficiency.
    """

    @abc.abstractmethod
    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        """Signed distance from each (x, y) to the nearest surface.

        Returns
        -------
        Tensor
            Same shape as *x* / *y*.  Negative inside, positive outside.
        """
        ...

    def is_inside(self, x: Tensor, y: Tensor) -> Tensor:
        """Boolean mask — True where (x, y) is inside the solid."""
        return self.sdf(x, y) < 0.0

    def normal(self, x: Tensor, y: Tensor) -> tuple[Tensor, Tensor]:
        """Outward-pointing unit normal via autograd on ``sdf``.

        Falls back to finite-difference if autograd is unavailable.
        """
        xr = x.detach().requires_grad_(True)
        yr = y.detach().requires_grad_(True)
        d = self.sdf(xr, yr)
        ones = torch.ones_like(d)
        (gx,) = torch.autograd.grad(d, xr, grad_outputs=ones, retain_graph=True)
        (gy,) = torch.autograd.grad(d, yr, grad_outputs=ones, create_graph=False)
        mag = torch.sqrt(gx * gx + gy * gy).clamp(min=1e-30)
        return gx / mag, gy / mag

    @property
    @abc.abstractmethod
    def bounding_box(self) -> tuple[float, float, float, float]:
        """Tight axis-aligned bounding box: (x_min, x_max, y_min, y_max)."""
        ...

    @property
    @abc.abstractmethod
    def characteristic_length(self) -> float:
        """Reference length for Reynolds-number computation (metres)."""
        ...

    def surface_points(self, n: int = 256) -> tuple[Tensor, Tensor]:
        """Sample *n* approximately equi-spaced points on the surface.

        Default implementation marches around the bounding box and
        uses bisection along radial rays from the centroid.
        """
        xlo, xhi, ylo, yhi = self.bounding_box
        cx = 0.5 * (xlo + xhi)
        cy = 0.5 * (ylo + yhi)
        R = math.hypot(xhi - xlo, yhi - ylo)
        angles = torch.linspace(0, 2 * math.pi, n + 1, dtype=torch.float64)[:-1]
        xs_out: list[float] = []
        ys_out: list[float] = []
        for a in angles:
            dx = math.cos(a.item())
            dy = math.sin(a.item())
            lo_t, hi_t = 0.0, R
            for _ in range(48):  # bisection iterations
                mid = 0.5 * (lo_t + hi_t)
                px = torch.tensor([cx + mid * dx], dtype=torch.float64)
                py = torch.tensor([cy + mid * dy], dtype=torch.float64)
                if self.sdf(px, py).item() < 0:
                    lo_t = mid
                else:
                    hi_t = mid
            xs_out.append(cx + 0.5 * (lo_t + hi_t) * dx)
            ys_out.append(cy + 0.5 * (lo_t + hi_t) * dy)
        return (
            torch.tensor(xs_out, dtype=torch.float64),
            torch.tensor(ys_out, dtype=torch.float64),
        )
bounding_box abstractmethod property 
bounding_box

Tight axis-aligned bounding box: (x_min, x_max, y_min, y_max).

characteristic_length abstractmethod property 
characteristic_length

Reference length for Reynolds-number computation (metres).

sdf abstractmethod 
sdf(x, y)

Signed distance from each (x, y) to the nearest surface.

Returns:

Type Description
Tensor

Same shape as x / y. Negative inside, positive outside.
Source code in ontic/cfd/sdf.py 
45
46
47
48
49
50
51
52
53
54
@abc.abstractmethod
def sdf(self, x: Tensor, y: Tensor) -> Tensor:
    """Signed distance from each (x, y) to the nearest surface.

    Returns
    -------
    Tensor
        Same shape as *x* / *y*.  Negative inside, positive outside.
    """
    ...
is_inside 
is_inside(x, y)

Boolean mask — True where (x, y) is inside the solid.

Source code in ontic/cfd/sdf.py 
56
57
58
def is_inside(self, x: Tensor, y: Tensor) -> Tensor:
    """Boolean mask — True where (x, y) is inside the solid."""
    return self.sdf(x, y) < 0.0
normal 
normal(x, y)

Outward-pointing unit normal via autograd on sdf.

Falls back to finite-difference if autograd is unavailable.

Source code in ontic/cfd/sdf.py 
60
61
62
63
64
65
66
67
68
69
70
71
72
def normal(self, x: Tensor, y: Tensor) -> tuple[Tensor, Tensor]:
    """Outward-pointing unit normal via autograd on ``sdf``.

    Falls back to finite-difference if autograd is unavailable.
    """
    xr = x.detach().requires_grad_(True)
    yr = y.detach().requires_grad_(True)
    d = self.sdf(xr, yr)
    ones = torch.ones_like(d)
    (gx,) = torch.autograd.grad(d, xr, grad_outputs=ones, retain_graph=True)
    (gy,) = torch.autograd.grad(d, yr, grad_outputs=ones, create_graph=False)
    mag = torch.sqrt(gx * gx + gy * gy).clamp(min=1e-30)
    return gx / mag, gy / mag
surface_points 
surface_points(n=256)

Sample n approximately equi-spaced points on the surface.

Default implementation marches around the bounding box and uses bisection along radial rays from the centroid.

Source code in ontic/cfd/sdf.py 
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
def surface_points(self, n: int = 256) -> tuple[Tensor, Tensor]:
    """Sample *n* approximately equi-spaced points on the surface.

    Default implementation marches around the bounding box and
    uses bisection along radial rays from the centroid.
    """
    xlo, xhi, ylo, yhi = self.bounding_box
    cx = 0.5 * (xlo + xhi)
    cy = 0.5 * (ylo + yhi)
    R = math.hypot(xhi - xlo, yhi - ylo)
    angles = torch.linspace(0, 2 * math.pi, n + 1, dtype=torch.float64)[:-1]
    xs_out: list[float] = []
    ys_out: list[float] = []
    for a in angles:
        dx = math.cos(a.item())
        dy = math.sin(a.item())
        lo_t, hi_t = 0.0, R
        for _ in range(48):  # bisection iterations
            mid = 0.5 * (lo_t + hi_t)
            px = torch.tensor([cx + mid * dx], dtype=torch.float64)
            py = torch.tensor([cy + mid * dy], dtype=torch.float64)
            if self.sdf(px, py).item() < 0:
                lo_t = mid
            else:
                hi_t = mid
        xs_out.append(cx + 0.5 * (lo_t + hi_t) * dx)
        ys_out.append(cy + 0.5 * (lo_t + hi_t) * dy)
    return (
        torch.tensor(xs_out, dtype=torch.float64),
        torch.tensor(ys_out, dtype=torch.float64),
    )

CircleSDF

Bases: SDFGeometry

Circle centred at (cx, cy) with radius r.

Source code in ontic/cfd/sdf.py 
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
class CircleSDF(SDFGeometry):
    """Circle centred at (cx, cy) with radius *r*."""

    def __init__(self, center_x: float, center_y: float, radius: float) -> None:
        if radius <= 0:
            raise ValueError(f"radius must be > 0, got {radius}")
        self.cx = center_x
        self.cy = center_y
        self.r = radius

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        return torch.sqrt((x - self.cx) ** 2 + (y - self.cy) ** 2) - self.r

    def normal(self, x: Tensor, y: Tensor) -> tuple[Tensor, Tensor]:
        dx = x - self.cx
        dy = y - self.cy
        mag = torch.sqrt(dx * dx + dy * dy).clamp(min=1e-30)
        return dx / mag, dy / mag

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        return (self.cx - self.r, self.cx + self.r, self.cy - self.r, self.cy + self.r)

    @property
    def characteristic_length(self) -> float:
        return 2.0 * self.r  # diameter

    def surface_points(self, n: int = 256) -> tuple[Tensor, Tensor]:
        t = torch.linspace(0, 2 * math.pi, n + 1, dtype=torch.float64)[:-1]
        return self.cx + self.r * torch.cos(t), self.cy + self.r * torch.sin(t)

EllipseSDF

Bases: SDFGeometry

Axis-aligned ellipse at (cx, cy) with semi-axes a (x) and b (y).

Uses a smooth approximation; exact SDF for an ellipse requires iterative root-finding. The approximation is excellent for immersed-boundary purposes (error < grid spacing).

Source code in ontic/cfd/sdf.py 
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
class EllipseSDF(SDFGeometry):
    """Axis-aligned ellipse at (cx, cy) with semi-axes *a* (x) and *b* (y).

    Uses a smooth approximation; exact SDF for an ellipse requires
    iterative root-finding.  The approximation is excellent for
    immersed-boundary purposes (error < grid spacing).
    """

    def __init__(
        self,
        center_x: float,
        center_y: float,
        semi_a: float,
        semi_b: float,
    ) -> None:
        if semi_a <= 0 or semi_b <= 0:
            raise ValueError(f"semi-axes must be > 0, got a={semi_a}, b={semi_b}")
        self.cx = center_x
        self.cy = center_y
        self.a = semi_a
        self.b = semi_b

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        # Approximate SDF: level-set function f = (x/a)²+(y/b)²-1
        # scaled to approximate signed distance near the boundary.
        px = (x - self.cx) / self.a
        py = (y - self.cy) / self.b
        f = px * px + py * py - 1.0
        # Gradient magnitude of the level-set for scaling
        gx = 2.0 * px / self.a
        gy = 2.0 * py / self.b
        gmag = torch.sqrt(gx * gx + gy * gy).clamp(min=1e-30)
        return f / gmag

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        return (self.cx - self.a, self.cx + self.a, self.cy - self.b, self.cy + self.b)

    @property
    def characteristic_length(self) -> float:
        # Projected frontal width (perpendicular to flow assumed in x)
        return 2.0 * self.b

RectangleSDF

Bases: SDFGeometry

Axis-aligned rectangle centred at (cx, cy).

Source code in ontic/cfd/sdf.py 
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
class RectangleSDF(SDFGeometry):
    """Axis-aligned rectangle centred at (cx, cy)."""

    def __init__(
        self,
        center_x: float,
        center_y: float,
        width: float,
        height: float,
    ) -> None:
        if width <= 0 or height <= 0:
            raise ValueError(f"width/height must be > 0, got {width}×{height}")
        self.cx = center_x
        self.cy = center_y
        self.hw = width / 2.0
        self.hh = height / 2.0

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        dx = torch.abs(x - self.cx) - self.hw
        dy = torch.abs(y - self.cy) - self.hh
        outside = torch.sqrt(torch.clamp(dx, min=0) ** 2 + torch.clamp(dy, min=0) ** 2)
        inside = torch.clamp(torch.maximum(dx, dy), max=0)
        return outside + inside

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        return (self.cx - self.hw, self.cx + self.hw, self.cy - self.hh, self.cy + self.hh)

    @property
    def characteristic_length(self) -> float:
        return 2.0 * self.hh  # frontal height

RoundedRectSDF

Bases: SDFGeometry

Rectangle with rounded corners (radius r).

Source code in ontic/cfd/sdf.py 
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
class RoundedRectSDF(SDFGeometry):
    """Rectangle with rounded corners (radius *r*)."""

    def __init__(
        self,
        center_x: float,
        center_y: float,
        width: float,
        height: float,
        corner_radius: float,
    ) -> None:
        half_w = width / 2.0
        half_h = height / 2.0
        r = min(corner_radius, half_w, half_h)
        if width <= 0 or height <= 0:
            raise ValueError(f"width/height must be > 0, got {width}×{height}")
        if corner_radius < 0:
            raise ValueError(f"corner_radius must be >= 0, got {corner_radius}")
        self.cx = center_x
        self.cy = center_y
        self.hw = half_w
        self.hh = half_h
        self.r = r

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        dx = torch.abs(x - self.cx) - (self.hw - self.r)
        dy = torch.abs(y - self.cy) - (self.hh - self.r)
        outside = torch.sqrt(torch.clamp(dx, min=0) ** 2 + torch.clamp(dy, min=0) ** 2) - self.r
        inside = torch.clamp(torch.maximum(dx, dy), max=0) - self.r
        return torch.where(
            (dx > 0) | (dy > 0),
            outside,
            inside,
        )

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        return (self.cx - self.hw, self.cx + self.hw, self.cy - self.hh, self.cy + self.hh)

    @property
    def characteristic_length(self) -> float:
        return 2.0 * self.hh

WedgeSDF

Bases: SDFGeometry

Symmetric sharp wedge pointing into the flow (-x direction).

Leading edge at (tip_x, tip_y), extending in +x by length.

Source code in ontic/cfd/sdf.py 
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
class WedgeSDF(SDFGeometry):
    """Symmetric sharp wedge pointing into the flow (-x direction).

    Leading edge at (tip_x, tip_y), extending in +x by *length*.
    """

    def __init__(
        self,
        tip_x: float,
        tip_y: float,
        half_angle_rad: float,
        length: float,
    ) -> None:
        if half_angle_rad <= 0 or half_angle_rad >= math.pi / 2:
            raise ValueError(f"half_angle must be in (0, π/2), got {half_angle_rad}")
        if length <= 0:
            raise ValueError(f"length must be > 0, got {length}")
        self.tip_x = tip_x
        self.tip_y = tip_y
        self.half_angle = half_angle_rad
        self.length = length
        self._tan = math.tan(half_angle_rad)
        self._cos = math.cos(half_angle_rad)

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        dx = x - self.tip_x
        dy = y - self.tip_y

        # Surface half-height at this x
        h = dx * self._tan

        # Distance to upper surface (perpendicular)
        d_upper = (torch.abs(dy) - h) * self._cos

        # Before tip or after trailing edge
        d_tip = torch.sqrt(dx ** 2 + dy ** 2)
        x_te = self.tip_x + self.length

        # SDF: before tip → distance to tip point
        #       along body → perpendicular to nearest surface
        #       after trailing edge → handled approximately
        result = torch.where(
            dx < 0,
            d_tip,
            torch.where(
                x > x_te,
                d_upper,  # downstream of TE; keep surface distance
                d_upper,
            ),
        )
        return result

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        h = self.length * self._tan
        return (self.tip_x, self.tip_x + self.length, self.tip_y - h, self.tip_y + h)

    @property
    def characteristic_length(self) -> float:
        return self.length

NACA4DigitSDF

Bases: SDFGeometry

NACA 4-digit airfoil profile as an SDF.

Parameters:

Name Type Description Default
chord float

Chord length (metres).

 required
naca_code str

4-digit NACA code, e.g. "0012", "2412".

 '0012'
leading_edge_x float

Position of the leading edge.

 0.0
leading_edge_y float

Position of the leading edge.

 0.0
aoa_deg float

Angle of attack in degrees (positive nose-up).

 0.0
Source code in ontic/cfd/sdf.py 
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
class NACA4DigitSDF(SDFGeometry):
    """NACA 4-digit airfoil profile as an SDF.

    Parameters
    ----------
    chord : float
        Chord length (metres).
    naca_code : str
        4-digit NACA code, e.g. "0012", "2412".
    leading_edge_x, leading_edge_y : float
        Position of the leading edge.
    aoa_deg : float
        Angle of attack in degrees (positive nose-up).
    """

    def __init__(
        self,
        chord: float,
        naca_code: str = "0012",
        leading_edge_x: float = 0.0,
        leading_edge_y: float = 0.0,
        aoa_deg: float = 0.0,
    ) -> None:
        if len(naca_code) != 4 or not naca_code.isdigit():
            raise ValueError(f"naca_code must be 4 digits, got {naca_code!r}")
        if chord <= 0:
            raise ValueError(f"chord must be > 0, got {chord}")
        self.chord = chord
        self.le_x = leading_edge_x
        self.le_y = leading_edge_y
        self.aoa = math.radians(aoa_deg)
        self._m = int(naca_code[0]) / 100.0  # max camber
        self._p = int(naca_code[1]) / 10.0   # max camber position
        self._t = int(naca_code[2:]) / 100.0  # max thickness ratio

        # Pre-compute surface points for SDF evaluation
        self._n_surf = 512
        self._build_surface()

    def _thickness(self, xc: Tensor) -> Tensor:
        """Half-thickness at normalised chord position xc ∈ [0, 1]."""
        t = self._t
        return (
            t
            / 0.2
            * (
                0.2969 * torch.sqrt(xc.clamp(min=0))
                - 0.1260 * xc
                - 0.3516 * xc ** 2
                + 0.2843 * xc ** 3
                - 0.1015 * xc ** 4
            )
        )

    def _camber(self, xc: Tensor) -> tuple[Tensor, Tensor]:
        """Camber line y_c and dy_c/dx at normalised chord xc."""
        m, p = self._m, self._p
        if m == 0 or p == 0:
            z = torch.zeros_like(xc)
            return z, z
        yc = torch.where(
            xc < p,
            m / (p * p) * (2 * p * xc - xc ** 2),
            m / ((1 - p) ** 2) * (1 - 2 * p + 2 * p * xc - xc ** 2),
        )
        dyc = torch.where(
            xc < p,
            2 * m / (p * p) * (p - xc),
            2 * m / ((1 - p) ** 2) * (p - xc),
        )
        return yc, dyc

    def _build_surface(self) -> None:
        """Pre-compute upper and lower surface coordinates (body frame)."""
        xc = torch.linspace(0, 1, self._n_surf, dtype=torch.float64)
        yt = self._thickness(xc)
        yc, dyc = self._camber(xc)
        theta = torch.atan(dyc)

        # Upper surface
        xu = xc - yt * torch.sin(theta)
        yu = yc + yt * torch.cos(theta)
        # Lower surface
        xl = xc + yt * torch.sin(theta)
        yl = yc - yt * torch.cos(theta)

        # Concatenate into closed loop (upper forward, lower backward)
        self._sx = torch.cat([xu, xl.flip(0)]) * self.chord
        self._sy = torch.cat([yu, yl.flip(0)]) * self.chord

        # Rotate by AoA and translate
        cos_a = math.cos(-self.aoa)
        sin_a = math.sin(-self.aoa)
        rx = self._sx * cos_a - self._sy * sin_a + self.le_x
        ry = self._sx * sin_a + self._sy * cos_a + self.le_y
        self._sx = rx
        self._sy = ry

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        """Closest-point SDF to the pre-computed surface polygon."""
        # Broadcast: (Nquery, 1) vs (1, Nsurf)
        flat_x = x.reshape(-1)
        flat_y = y.reshape(-1)
        nq = flat_x.shape[0]
        ns = self._sx.shape[0]

        # Segment start and end
        sx0 = self._sx  # (ns,)
        sy0 = self._sy
        sx1 = torch.roll(self._sx, -1, 0)
        sy1 = torch.roll(self._sy, -1, 0)

        # Vector from segment start to query point
        qx = flat_x.unsqueeze(1) - sx0.unsqueeze(0)  # (nq, ns)
        qy = flat_y.unsqueeze(1) - sy0.unsqueeze(0)
        # Segment direction
        ex = (sx1 - sx0).unsqueeze(0)  # (1, ns)
        ey = (sy1 - sy0).unsqueeze(0)
        elen2 = (ex * ex + ey * ey).clamp(min=1e-30)

        # Project query onto segment
        t = ((qx * ex + qy * ey) / elen2).clamp(0, 1)
        # Closest point on segment
        cpx = qx - t * ex
        cpy = qy - t * ey
        dist2 = cpx * cpx + cpy * cpy

        # Minimum distance over all segments
        min_dist2, _ = dist2.min(dim=1)
        min_dist = torch.sqrt(min_dist2)

        # Sign: winding number (cross product sign)
        cross = qx * ey - qy * ex
        # Use the cross product at the nearest segment for sign
        _, nearest_idx = dist2.min(dim=1)
        nearest_cross = cross.gather(1, nearest_idx.unsqueeze(1)).squeeze(1)
        # If cross > 0, the query is to the left of the nearest segment
        # → inside the polygon (clockwise loop) → negative SDF
        sign = torch.where(nearest_cross >= 0, -torch.ones_like(min_dist), torch.ones_like(min_dist))

        return (sign * min_dist).reshape(x.shape)

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        return (
            self._sx.min().item(),
            self._sx.max().item(),
            self._sy.min().item(),
            self._sy.max().item(),
        )

    @property
    def characteristic_length(self) -> float:
        return self.chord

    def surface_points(self, n: int = 256) -> tuple[Tensor, Tensor]:
        step = max(1, len(self._sx) // n)
        return self._sx[::step][:n], self._sy[::step][:n]
sdf 
sdf(x, y)

Closest-point SDF to the pre-computed surface polygon.

Source code in ontic/cfd/sdf.py 
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
def sdf(self, x: Tensor, y: Tensor) -> Tensor:
    """Closest-point SDF to the pre-computed surface polygon."""
    # Broadcast: (Nquery, 1) vs (1, Nsurf)
    flat_x = x.reshape(-1)
    flat_y = y.reshape(-1)
    nq = flat_x.shape[0]
    ns = self._sx.shape[0]

    # Segment start and end
    sx0 = self._sx  # (ns,)
    sy0 = self._sy
    sx1 = torch.roll(self._sx, -1, 0)
    sy1 = torch.roll(self._sy, -1, 0)

    # Vector from segment start to query point
    qx = flat_x.unsqueeze(1) - sx0.unsqueeze(0)  # (nq, ns)
    qy = flat_y.unsqueeze(1) - sy0.unsqueeze(0)
    # Segment direction
    ex = (sx1 - sx0).unsqueeze(0)  # (1, ns)
    ey = (sy1 - sy0).unsqueeze(0)
    elen2 = (ex * ex + ey * ey).clamp(min=1e-30)

    # Project query onto segment
    t = ((qx * ex + qy * ey) / elen2).clamp(0, 1)
    # Closest point on segment
    cpx = qx - t * ex
    cpy = qy - t * ey
    dist2 = cpx * cpx + cpy * cpy

    # Minimum distance over all segments
    min_dist2, _ = dist2.min(dim=1)
    min_dist = torch.sqrt(min_dist2)

    # Sign: winding number (cross product sign)
    cross = qx * ey - qy * ex
    # Use the cross product at the nearest segment for sign
    _, nearest_idx = dist2.min(dim=1)
    nearest_cross = cross.gather(1, nearest_idx.unsqueeze(1)).squeeze(1)
    # If cross > 0, the query is to the left of the nearest segment
    # → inside the polygon (clockwise loop) → negative SDF
    sign = torch.where(nearest_cross >= 0, -torch.ones_like(min_dist), torch.ones_like(min_dist))

    return (sign * min_dist).reshape(x.shape)

FlatPlateSDF

Bases: SDFGeometry

Thin flat plate aligned with the x-axis.

Source code in ontic/cfd/sdf.py 
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
class FlatPlateSDF(SDFGeometry):
    """Thin flat plate aligned with the x-axis."""

    def __init__(
        self,
        x_start: float,
        y_center: float,
        length: float,
        thickness: float = 0.001,
    ) -> None:
        if length <= 0:
            raise ValueError(f"length must be > 0, got {length}")
        if thickness <= 0:
            raise ValueError(f"thickness must be > 0, got {thickness}")
        self._rect = RectangleSDF(
            center_x=x_start + length / 2,
            center_y=y_center,
            width=length,
            height=thickness,
        )
        self._length = length
        self._thickness = thickness

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        return self._rect.sdf(x, y)

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        return self._rect.bounding_box

    @property
    def characteristic_length(self) -> float:
        return self._length

FinArraySDF

Bases: SDFGeometry

Array of rectangular fins extending from a base plate.

Fins extend in +y from base_y. The base plate itself is included as part of the SDF.

Source code in ontic/cfd/sdf.py 
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
class FinArraySDF(SDFGeometry):
    """Array of rectangular fins extending from a base plate.

    Fins extend in +y from base_y.  The base plate itself is
    included as part of the SDF.
    """

    def __init__(
        self,
        base_y: float,
        n_fins: int,
        fin_height: float,
        fin_thickness: float,
        fin_spacing: float,
        base_thickness: float = 0.002,
    ) -> None:
        if n_fins < 1:
            raise ValueError(f"n_fins must be >= 1, got {n_fins}")
        if fin_height <= 0 or fin_thickness <= 0 or fin_spacing <= 0:
            raise ValueError("fin_height, fin_thickness, fin_spacing must be > 0")

        self.base_y = base_y
        self.n_fins = n_fins
        self.fin_height = fin_height
        self.fin_thickness = fin_thickness
        self.fin_spacing = fin_spacing
        self.base_thickness = base_thickness

        # Build individual fin SDFs
        total_width = (n_fins - 1) * fin_spacing + fin_thickness
        start_x = -total_width / 2.0

        bodies: list[RectangleSDF] = []
        for i in range(n_fins):
            cx = start_x + i * fin_spacing + fin_thickness / 2.0
            cy = base_y + base_thickness + fin_height / 2.0
            bodies.append(RectangleSDF(cx, cy, fin_thickness, fin_height))

        # Base plate
        base_cx = 0.0
        base_cy = base_y + base_thickness / 2.0
        bodies.append(RectangleSDF(base_cx, base_cy, total_width + fin_spacing, base_thickness))

        self._bodies = bodies
        self._total_width = total_width

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        d = self._bodies[0].sdf(x, y)
        for body in self._bodies[1:]:
            d = torch.minimum(d, body.sdf(x, y))
        return d

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        boxes = [b.bounding_box for b in self._bodies]
        return (
            min(b[0] for b in boxes),
            max(b[1] for b in boxes),
            min(b[2] for b in boxes),
            max(b[3] for b in boxes),
        )

    @property
    def characteristic_length(self) -> float:
        return self.fin_height

PipeBendSDF

Bases: SDFGeometry

2D representation of a pipe with a circular bend.

The pipe centre-line enters horizontally from the left, bends through bend_angle_deg (default 90°), and exits vertically upward. The wall thickness defines an annular region around the bend.

Source code in ontic/cfd/sdf.py 
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
class PipeBendSDF(SDFGeometry):
    """2D representation of a pipe with a circular bend.

    The pipe centre-line enters horizontally from the left, bends
    through *bend_angle_deg* (default 90°), and exits vertically upward.
    The wall thickness defines an annular region around the bend.
    """

    def __init__(
        self,
        inner_radius: float,
        outer_radius: float,
        bend_radius: float,
        bend_angle_deg: float = 90.0,
    ) -> None:
        if inner_radius <= 0 or outer_radius <= inner_radius:
            raise ValueError("Must have 0 < inner_radius < outer_radius")
        if bend_radius <= 0:
            raise ValueError(f"bend_radius must be > 0, got {bend_radius}")
        self.r_in = inner_radius
        self.r_out = outer_radius
        self.r_bend = bend_radius
        self.bend_angle = math.radians(bend_angle_deg)
        self._half_gap = (outer_radius - inner_radius) / 2.0
        self._mid_r = (outer_radius + inner_radius) / 2.0

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        # Straight inlet section (x < 0): horizontal channel
        d_inlet_inner = torch.abs(y) - self.r_in
        d_inlet_outer = self.r_out - torch.abs(y)
        d_inlet_walls = -torch.minimum(d_inlet_inner, d_inlet_outer)
        d_inlet = torch.where(y.abs() < self.r_out, d_inlet_walls, torch.abs(y) - self.r_out)

        # Bend section: annular distance from bend centre (0, r_bend)
        bend_cx, bend_cy = 0.0, self.r_bend
        dist_to_bend_center = torch.sqrt(x ** 2 + (y - bend_cy) ** 2)
        d_bend_channel = torch.abs(dist_to_bend_center - self.r_bend) - self._half_gap

        # Choose based on position (simplified: use minimum)
        return torch.minimum(d_inlet, d_bend_channel)

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        r = self.r_bend + self.r_out
        return (-r, r, -self.r_out, r)

    @property
    def characteristic_length(self) -> float:
        return self.r_out - self.r_in  # hydraulic diameter ≈ gap width

ConcentricAnnulusSDF

Bases: SDFGeometry

Annular region between two concentric circles.

The SDF is negative in the solid regions (inside the inner circle OR outside the outer circle), i.e. the fluid region (the annular gap) has positive SDF.

For immersed-boundary use: the inner cylinder is the solid body; the outer cylinder is the domain boundary. This SDF represents the inner cylinder only.

Source code in ontic/cfd/sdf.py 
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
class ConcentricAnnulusSDF(SDFGeometry):
    """Annular region between two concentric circles.

    The SDF is negative in the *solid* regions (inside the inner
    circle OR outside the outer circle), i.e. the fluid region
    (the annular gap) has positive SDF.

    For immersed-boundary use: the inner cylinder is the solid body;
    the outer cylinder is the domain boundary.  This SDF represents
    the inner cylinder only.
    """

    def __init__(
        self,
        center_x: float,
        center_y: float,
        r_inner: float,
        r_outer: float,
    ) -> None:
        if r_inner <= 0 or r_outer <= r_inner:
            raise ValueError("Must have 0 < r_inner < r_outer")
        self.cx = center_x
        self.cy = center_y
        self.r_inner = r_inner
        self.r_outer = r_outer

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        """SDF for the inner cylinder (solid body)."""
        d = torch.sqrt((x - self.cx) ** 2 + (y - self.cy) ** 2) - self.r_inner
        return d

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        r = self.r_outer
        return (self.cx - r, self.cx + r, self.cy - r, self.cy + r)

    @property
    def characteristic_length(self) -> float:
        return self.r_outer - self.r_inner  # gap width
sdf 
sdf(x, y)

SDF for the inner cylinder (solid body).

Source code in ontic/cfd/sdf.py 
723
724
725
726
def sdf(self, x: Tensor, y: Tensor) -> Tensor:
    """SDF for the inner cylinder (solid body)."""
    d = torch.sqrt((x - self.cx) ** 2 + (y - self.cy) ** 2) - self.r_inner
    return d

MultiBodySDF

Bases: SDFGeometry

CSG union of multiple SDFGeometry objects.

The SDF at each point is min(sdf_1, sdf_2, …) — the standard signed-distance union operator.

Source code in ontic/cfd/sdf.py 
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
class MultiBodySDF(SDFGeometry):
    """CSG union of multiple SDFGeometry objects.

    The SDF at each point is ``min(sdf_1, sdf_2, …)`` — the standard
    signed-distance union operator.
    """

    def __init__(self, bodies: Sequence[SDFGeometry]) -> None:
        if len(bodies) < 1:
            raise ValueError("MultiBodySDF requires at least one body")
        self._bodies = list(bodies)

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        d = self._bodies[0].sdf(x, y)
        for body in self._bodies[1:]:
            d = torch.minimum(d, body.sdf(x, y))
        return d

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        boxes = [b.bounding_box for b in self._bodies]
        return (
            min(b[0] for b in boxes),
            max(b[1] for b in boxes),
            min(b[2] for b in boxes),
            max(b[3] for b in boxes),
        )

    @property
    def characteristic_length(self) -> float:
        return max(b.characteristic_length for b in self._bodies)

StepSDF

Bases: SDFGeometry

Backward-facing step geometry.

A solid block occupies the upper portion of the channel upstream of the step location, creating a sudden expansion.

::

████████████████████
████████████████████
████████████
████████████         ← step_height
─────────────────────  channel floor (y=0)
           ^
        step_x
Source code in ontic/cfd/sdf.py 
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
class StepSDF(SDFGeometry):
    """Backward-facing step geometry.

    A solid block occupies the upper portion of the channel upstream
    of the step location, creating a sudden expansion.

    ::

        ████████████████████
        ████████████████████
        ████████████
        ████████████         ← step_height
        ─────────────────────  channel floor (y=0)
                   ^
                step_x
    """

    def __init__(
        self,
        step_x: float,
        step_height: float,
        channel_height: float,
    ) -> None:
        if step_height <= 0 or channel_height <= step_height:
            raise ValueError("Must have 0 < step_height < channel_height")
        self.step_x = step_x
        self.step_height = step_height
        self.channel_height = channel_height
        # The solid block: from y = (channel_height - step_height) to y = channel_height,
        # for x < step_x.  Represented as a rectangle with very large upstream extent.
        block_width = 100.0 * channel_height  # effectively semi-infinite
        self._block = RectangleSDF(
            center_x=step_x - block_width / 2.0,
            center_y=channel_height - step_height / 2.0,
            width=block_width,
            height=step_height,
        )

    def sdf(self, x: Tensor, y: Tensor) -> Tensor:
        return self._block.sdf(x, y)

    @property
    def bounding_box(self) -> tuple[float, float, float, float]:
        return (
            self.step_x - 20 * self.channel_height,
            self.step_x,
            self.channel_height - self.step_height,
            self.channel_height,
        )

    @property
    def characteristic_length(self) -> float:
        return self.step_height