Numerical results for the unit sphere are shown in Figure 2. As predicted in (7), the error in V ( Δ x ) is approximately proportional to ( Δ x ) 2 . The error in V extrap ( Δ x ) is significantly below that in V ( Δ x ) , demonstrating the benefit of extrapolation. Note that the extrapolation component does not require additional data or information on function f ( x ) . V ( 2 Δ x ) in the extrapolation is obtained from down-sampling the given discrete representation of f ( x ) .

(a) (b)

Figure 2. Left: Triangulation of the unit sphere as an isosurface of function f ( x ) given in (11). Right: errors in numerical volumes V ( Δ x ) and V extrap ( Δ x ) vs mesh spacing Δ x . Dashed line is ( Δ x ) 2 vs Δ x serving as a visual guide.

In Figure 3 for the axis-aligned ellipsoid, the volume computation methods show the same behavior as in Figure 2. The error in V ( Δ x ) is approximately proportional to ( Δ x ) 2 ; the error in V extrap ( Δ x ) is significantly smaller than that in V ( Δ x ) , confirming the second-order accuracy of triangulation and the accuracy improvement of extrapolation.

(a) (b)

Figure 3. Left: Triangulation of the axis-aligned ellipsoid as an isosurface of f ( x ) in (12). Right: errors in numerical volumes V ( Δ x ) and V extrap ( Δ x ) vs mesh spacing Δ x .

In (13), matrix A is orthogonal, satisfying A T A = I . Geometrically, multiplying by A corresponds to rotating the 3D region with respect to the z-axis by 45˚ and then rotating it with respect to the y-axis by (−30˚). The enclosed region in (13) is congruent to that of Subsection 3.2 via an orthogonal transformation, which preserves the analytical volume.

The discrete representation of function f ( x ) on a fixed 3D grid, however, varies with the orientation. Accordingly, the corresponding numerical volume based on the discrete representation is expected to be affected. In this sense, the rotated ellipsoid provides a completely new test on the volume computation methods, although the exact volume stays the same. For the rotated ellipsoid, in Figure 4, we observe the same behavior as in Figure 2 and Figure 3. The error in V ( Δ x ) is approximately proportional to ( Δ x ) 2 ; the error in V extrap ( Δ x ) is much lower than that in V ( Δ x ) .

(a) (b)

Figure 4. Left: Triangulation of the rotated ellipsoid as an isosurface of f ( x ) in (13). Right: errors in numerical volumes V ( Δ x ) and V extrap ( Δ x ) vs mesh spacing Δ x .

where β ( a , b ) is the beta function defined as

The beta function is available in standard software packages.

The regular ellipsoid (12) is a special case of the superellipsoid (14) at ε 1 = ε 2 = 1 . In the case of ε 1 = ε 2 = 2 n , the enclosed region is described by

As n → + ∞ , the enclosed region converges to the 3D rectangular prism:

For ε 1 = ε 2 < 1 , the enclosed superellipsoid is between the regular ellipsoid and the 3D rectangular prism. Indeed, the principal sides of the superellipsoid shown in the left panel of Figure 5 are flatter than those of the regular ellipsoid in Figure 3. For the superellipsoid, even with this shape change in the enclosed region, the error trends in volume computation remain the same as in Figures 2-4. The error in V ( Δ x ) is proportional to ( Δ x ) 2 ; the error in V extrap ( Δ x ) is significantly smaller than that in V ( Δ x ) .

(a) (b)

Figure 5. Left: Triangulation of the axis-aligned superellipsoid as an isosurface of f ( x ) in (14). Right: errors in numerical volumes V ( Δ x ) and V extrap ( Δ x ) vs mesh spacing Δ x .

The enclosed region in (15) is congruent to that in (14) of Subsection 3.4 via an orthogonal transformation. In (15), the orthogonal transformation represented by matrix A is the same as that in (13). Similar to what we observed in Figures 2-5, Figure 6 shows that the errors in V ( Δ x ) is proportional to ( Δ x ) 2 ; the error in V extrap ( Δ x ) is significantly smaller than that in V ( Δ x ) , demonstrating the steady improvement of extrapolation over several test problems.

(a) (b)

Figure 6. Left: Triangulation of the rotated superellipsoid as an isosurface of f ( x ) in (15). Right: errors in numerical volumes V ( Δ x ) and V extrap ( Δ x ) vs mesh spacing Δ x .

A triangulation of the axis-aligned elliptic ring torus is shown in the left panel of Figure 7. Geometrically, the centerline of the torus is the circle of radius R centered at the origin in the ( x , y ) -plane. Perpendicular to the centerline, the cross section of the torus is an ellipse with the semi minor axis r in the horizontal direction and the semi major axis ( c r ) in the z-direction. For the elliptic ring torus in (16), c > 1 . The exact volume of the enclosed region is given by the product of the center line arclength ( 2 π R ) and the cross-sectional area ( π c r 2 ). We use the ring torus as a test problem to examine the error trends in numerical volumes.

(a) (b)

Figure 7. Left: Triangulation of the axis-aligned elliptic ring torus as an isosurface of f ( x ) in (16). Right: errors in numerical volumes V ( Δ x ) and V extrap ( Δ x ) vs mesh spacing Δ x .

Note that numerical volumes are computed from a discrete representation of function f ( x ) on a 3D grid of mesh spacing Δ x , not using the full analytical expression of f ( x ) . The exact volume is not used in computing numerical volumes; it is used only in evaluating the errors of numerical volumes. The right panel of Figure 7 shows that the errors in V ( Δ x ) and V extrap ( Δ x ) both decrease with Δ x with the latter significantly below the former.

In (17), the orthogonal transformation represented by matrix B is different from that by matrix A in (13) and (15). If we first rotate the elliptic ring torus about the z-axis, due to the axial symmetry of the object, this rotation would not change the object at all. Instead, in (17), the transformation represented by matrix B corresponds to rotating the 3D object first with respect to the y-axis by (−30˚) and then rotating it with respect to the z-axis by 45˚. A triangulation of the rotated elliptic ring torus is shown in the left panel of Figure 8. The error trends of numerical volumes are plotted in the right panel of Figure 8. Similar to what we observed in all other test problems, the error in V ( Δ x ) is proportional to ( Δ x ) 2 ; the error in V extrap ( Δ x ) is significantly below that in V ( Δ x ) .

We introduce the physical quantities and equations used in our model.

T ( x , y , z , t ) is the skin temperature at position ( x , y , z ) at time t . ρ m is the mass density, C p the specific heat capacity, and k the heat conductivity of skin. In this study, we consider a single layer of uniform skin where all material properties are independent of position ( x , y , z ) .The energy balance gives the governing equation for T ( x , y , z , t ) [13]:

μ is the absorption coefficient of skin for the electromagnetic frequency as the wave propagates in the skin. μ describes the fraction of power absorbed per unit depth as the beam propagates in the skin, which is a lossy medium. The remaining power density at depth z is governed by the Beer-Lambert law [14].

μ has the physical dimension of 1/[length]. ( 1 / μ ) gives the characteristic penetration depth of the electromagnetic frequency, the depth at which the power density is attenuated by a factor of 1 / e . ( 1 / μ ) serves as the depth scale in non-dimensionalization.

P d ( x , y ) is the power density passing through the surface into skin tissue. Since skin is a lossy medium ( μ > 0 ), as described in Beer-Lambert law in (19), the remaining power density at depth z is virtually zero at large ( μ z ). All of P d ( x , y ) is converted to heat during the electromagnetic propagation in the skin. The heat production rate per volume at depth z is P d ( x , y ) μ e − μ z , which is the heat source in the skin temperature evolution governed by the 3D heat equation (18).In our simple model, the power density P d ( x , y ) is an axially symmetric Gaussian.

where w is the standard beam radius defined as the distance from the beam center to the location where the power density drops to 1 / e 2 ≈ 13.5 % of the peak value; σ is the RMS (root mean square) beam radius, which is the standard deviation of the Gaussian distribution. These two are related by σ = w / 2 .

In the z-direction, the characteristic scale is given by the penetration depth z s = 1 / μ . In the ( x , y ) -lateral directions, the characteristic scale is given by the RMS beam radius r s = σ . In the time direction, the characteristic scale is given by the quantity t s ≡ ρ m C p k μ 2 . With these scales, we carry out non-dimensionalization and recycle the same notations for all non-dimensional functions and variables afterward. The resulting non-dimensional governing equation is:

where ε ≡ z s r s is the ratio of the penetration depth to the lateral scale [15].

In electromagnetic heating applications using millimeter wavelength, the electromagnetic penetration depth is sub-millimeter while the RMS radius of the Gaussian beam is several centimeters [16]. In these applications, the depth-to-lateral-length ratio satisfies ε ≪ 1 . Thus, the effect of the lateral heat conduction ε 2 ( ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 ) is negligible in comparison with other terms in (20). To the leading order, the governing equation becomes:

We continue discussing the simplified system governing the skin temperature evolution.

Before the start of electromagnetic exposure, the skin is in its thermal homeostasis. The homeostatic temperature varies gradually from the core temperature inside skin tissue to a lower temperature at the skin surface. In our simplified model, however, we neglect this spatial variation in homeostatic temperature and assume that skin baseline temperature is a uniform constant T base ( phy ) at the beginning of electromagnetic exposure. This assumption is justified for millimeter wave exposure because the heating is concentrated in a very thin skin layer in which the variation of homeostatic temperature is small. This assumption leads to a simple initial condition.

In non-dimensionalization (20), physical temperature T ( phy ) is shifted by T base ( phy ) and normalized by the temperature scale Δ T ≡ T act ( phy ) − T base ( phy ) where T act ( phy ) is the physical temperature threshold for activating skin thermal nociceptors. Given the physical baseline temperature T base ( phy ) , the temperature scale Δ T is the temperature increase needed for skin thermal nociceptor activation. After normalization, the non-dimensional versions of baseline temperature and temperature threshold are

In homeostasis, the skin loses heat to the environment at its surface via blackbody radiation and convective cooling. However, the rate of heat loss from the skin surface is low. In a high-intensity, short-duration electromagnetic exposure, the rate of electromagnetic energy entering the skin tissue is much greater than the rate of slow heat loss at the skin surface. Thus, we neglect the slow heat loss at the skin surface and assume no heat exchange with the environment there. This assumption leads to a simple boundary condition.

The governing system for the evolution of skin temperature consists of a simplified non-dimensional equation and initial and boundary conditions.

Note that (22) has three key features: 1) it is linear; 2) ( x , y ) are not involve differential variables; and 3) the source term has separate dependences on ( x , y ) and on z . These features imply that the skin temperature T ( x , y , z , t ) has the form

where U ( z , t ) is a function of ( z , t ) only and satisfies

The solution of IBVP (24) has a closed-form analytical expression [17].

where erfc ( s ) is the complementary error function.

In this subsection, all quantities are non-dimensional. The activated skin region D at time t is enclosed by the isosurface of skin temperature T ( x , y , z , t ) at value T act = 1 . Using the expression of T ( x , y , z , t ) in (23), we write the activated skin region D as

In this simplified electromagnetic heating problem, the activated region D is axially symmetric. In more realistic electromagnetic heating applications, the electromagnetic beam may be moving relative to the skin, the incident angle may be oblique, or the skin material properties may be inhomogeneous. Under these complex conditions, we no longer have a closed-form analytical solution for skin temperature, which needs to be solved numerically on a 3D grid and in general is not axially symmetric [18][19]. Our volume computation methods are well capable of accommodating this complex situation. It requires only a discrete representation of skin temperature on a 3D grid; no symmetry is assumed. Here we use (26) to construct a non-symmetric problem for testing the accuracy of volume computation. We first find the activated volume of (26). The activated skin region of (26) is

where z L ( t ) ≡ U − 1 ( 1 P d ( 0 ) , t ) and U − 1 ( ⋅ , t ) is the inverse of z ↦ U ( z , t ) at fixed t . We express the activated skin volume in terms of a single integral.

We solve this integral accurately to machine precision using the composite Simpson’s rule, and use the result to assess the errors in numerical volumes (6) and (9). In the numerical test, we use P d ( 0 ) = 2.5 and t = 1 . To test the robustness of the volume computation, we shift the region in the x-direction by a z-dependent amount. This transformation destroys the axial symmetry while preserving the true volume, which is needed in error assessment. We set the test problem as follows.

Figure 10 shows a triangulation of the isosurface at the activation temperature threshold (left panel) and the errors in numerical volumes (right panel). The triangulation is based on a coarse grid with Δ x = 1 8 for illustrative purposes. The activated skin region D is enclosed by the isosurface and the skin surface ( z = 0 ). With the z-dependent shift in x , neither function f ( x ) in (28) nor the region D enclosed by the isosurface is axially symmetric. The volume computation procedure, however, is not impacted at all by the transformation. It uses only a discretization of function f ( x ) on a 3D numerical grid. The errors in numerical volumes behave as expected. The error in V ( Δ x ) is proportional to ( Δ x ) 2 ; the error in V extrap ( Δ x ) is significantly below that in V ( Δ x ) .