Topological Derivative: Theory & Applications

• Topological Derivative is a sensitivity measure that quantifies how an objective functional reacts to an infinitesimal perturbation of the domain’s topology.
• It is applied in electromagnetic imaging, structural design, and topology optimization to detect inclusions, cracks, and inhomogeneities via non-iterative reconstruction.
• Numerical studies confirm that normalization and multi-frequency averaging enhance imaging accuracy and robustly suppress artifacts.

A topological derivative quantifies the first-order sensitivity of a cost or objective functional with respect to an infinitesimal perturbation of the domain's topology. In practical terms, it measures how the value of a functional changes when a tiny inclusion (such as a hole or material inhomogeneity) is nucleated at a given point. This concept offers a principled mechanism for reconstructing shapes, identifying inhomogeneities, or optimizing material distributions based on PDE-constrained functionals in fields ranging from electromagnetic imaging to structural and electromagnetic design.

1. Mathematical Definition and Fundamental Role

The topological derivative $D_T \mathcal{J}(z)$ of a functional $\mathcal{J}$ is defined through the leading-order term in an asymptotic expansion involving the insertion of a vanishingly small perturbation (e.g., an inclusion at $z$) into the domain:

$$\mathcal{J}(\Omega_\varepsilon) = \mathcal{J}(\Omega) + f(\varepsilon) D_T \mathcal{J}(z) + o(f(\varepsilon)), \quad \varepsilon \to 0,$$

where $$\Omega_\varepsilon = \Omega \setminus B_\varepsilon(z)$$ or $\Omega \cup B_\varepsilon(z)$ (depending on context), and $f(\varepsilon)$ is the volume or area scaling with $\varepsilon$. For PDE-constrained problems, $D_T\mathcal{J}(z)$ is derived by differentiating the functional with respect to this perturbation, often via adjoint variables and asymptotic analysis.

Topological derivatives provide a universal sensitivity map over the domain, indicating where nucleation or removal of material would reduce or increase the objective. Unlike boundary shape derivatives, topological derivatives are defined throughout the domain, enabling detection of new topological features such as holes, cracks, or inclusions.

2. Analytical Formulation in Inverse and Direct Problems

For electromagnetic inverse scattering, the topological derivative is formulated with respect to an energy-based misfit between measured and simulated (background) fields. For example, in "Multi-frequency topological derivative for approximate shape acquisition of curve-like thin electromagnetic inhomogeneities" (Park, 2012), the functional is

$$E(\Omega; \omega) = \frac{1}{2} \sum_{l}\|u_{\text{tot}}^{(l)}(\cdot ; \omega) - u_{\text{bac}}^{(l)}(\cdot;\omega)\|^2$$

and the topological derivative at $z$ is

$$d_{TE}(z;\omega) = \Re \sum_{l} v_{\text{adj}}^{(l)}(z;\omega) \cdot \overline{u_{\text{bac}}^{(l)}(z;\omega)}$$

where $v_{\text{adj}}$ are adjoint solutions corresponding to boundary misfits. This structure generalizes across imaging and inverse problems, always relating the variation of the misfit to the state and adjoint fields.

In direct design problems, as in topology optimization of electric machines, the topological derivative expresses how introducing a hole (or new material) changes the objective. The asymptotic expansion involves the state and adjoint solution gradients at the perturbation point, possibly including polarization tensors for shape and conductivity contrasts (Gangl et al., 2013).

3. Multi-Frequency and Multi-Directional Analysis

Topological derivative–based imaging and optimization is often improved by aggregating data over multiple frequencies or directions. High frequencies resolve fine features but are sensitive to noise and oscillations; lower frequencies provide stability and better penetration. Averaging the normalized topological derivative over $K$ frequencies,

$$E(z; K) = \frac{1}{K} \sum_{k=1}^K E_{TD}(z; \omega_k)$$

reduces artifacts and improves robustness to noise and model uncertainties, as shown in electromagnetic imaging of thin inclusions (Park, 2012), and fast reconstruction of perfectly conducting cracks (Park, 2012).

Optimal configuration of incident directions is critical—only certain symmetric arrangements eliminate spurious oscillatory terms, a result revealed by Bessel function expansions of the imaging function (Park, 2017). For a small, symmetrically arranged set of incident directions, the infinite series of disturbance terms involving higher-order Bessel functions is effectively canceled, enabling robust imaging even with few measurements.

4. Normalization and Imaging Functional Structure

Normalization of the topological derivative is essential before aggregation. Each frequency's derivative is divided by its maximum over $z$:

$$E_{\mathrm{TD}}(z; \omega) = \frac{1}{2}\left( \frac{d_{T}\mathbb{E}_\varepsilon(z;\omega)}{\max d_{T}\mathbb{E}_\varepsilon(\cdot;\omega)} + \frac{d_{T}\mathbb{E}_\mu(z;\omega)}{\max d_{T}\mathbb{E}_\mu(\cdot;\omega)} \right)$$

This ensures uniform scaling between contributions (e.g., dielectric and magnetic), guarantees that the imaging map exhibits maximum values at the true inclusion location, and improves the interpretability of the sensitivity map. The normalized, multi-frequency imaging functional then highlights true support regions and suppresses noise-induced or oscillatory artifacts:

$$\mathbb{E}(z; K) = \frac{1}{K}\sum_{k=1}^K \mathbb{E}_{\mathrm{TD}}(z; \omega_k)$$

Theoretical analysis shows that, for thin inhomogeneities, the imaging function can be approximated by expressions involving Bessel functions such as $j_0(x) = \sin x / x$, ensuring sharp peaks at locations $z$ on the actual inclusion support.

5. Numerical Validation and Robustness

Extensive numerical studies demonstrate that topological derivative–based imaging is highly robust to noise—especially when multi-frequency averaging and normalization are applied (Park, 2012). Even with limited data (small numbers of incident directions or noisy boundary observations at $15\,\rm{dB}$ SNR), the imaging function reliably detects the true support of inclusions, crack locations, or conductivity anomalies.

In multi-inclusion and multi-contrast scenarios, the sensitivity map not only localizes inclusions but also reflects contrast-related amplitude variations. The derivative-based imaging function compares favorably with established non-iterative methods such as MUSIC and Kirchhoff migration, especially in regimes with few incident directions or significant noise (Park, 2012, Park, 2012).

Artifacts and spurious peaks are suppressed in the multi-frequency, normalized approach, producing imaging fields sharply concentrated at inclusion geometry. Where only a single frequency or non-optimal incident directions are used, spurious features and "ghost replicas" may occur, affirming the importance of frequency diversity and incident field design.

6. Applications, Limitations, and Extensions

Topological derivatives are deployed for:

• Non-iterative imaging of crack-like or thin inclusions in electromagnetic domains by mapping sensitivity fields;
• Fast localization of small conductivity anomalies for electrical impedance tomography;
• Providing initial guesses for iterative shape reconstruction algorithms (e.g., level-set methods);
• Robust initial design steps in topology optimization for electromagnetic and structural devices, where further iterative updates are guided by shape or material sensitivities.

While topological derivatives enable rapid, robust detection of small anomalies and inclusions, their spatial resolution is subject to the choice of frequency, number of incident directions, and noise levels. For multiple or extended anomalies, interference between defects may produce artifacts or neutralize the sensitivity, necessitating refined post-processing or iterative approaches.

7. Summary Table: Key Mathematical Structures

Context	Topological Derivative Formula	Application Example
Electromagnetic imaging (dielectric)	$$d_{TE_\varepsilon}(z;\omega)=\Re\sum_{l} v_\mathrm{adj}^{(l)}(z;\omega)\overline{u_\mathrm{bac}^{(l)}(z;\omega)}$$	Imaging thin inclusions (Park, 2012)
Electromagnetic imaging (magnetic)	$$d_{TE_\mu}(z;\omega)=\Re\sum_{l}\nabla v_\mathrm{adj}^{(l)}(z;\omega)\cdot\overline{\nabla u_\mathrm{bac}^{(l)}(z;\omega)}$$	(Park, 2012)
Multi-frequency imaging functional	$$\mathbb{E}(z;K)=\frac{1}{2K}\sum_k\left(\frac{d_{TE_\varepsilon}(z;\omega_k)}{\max d_{TE_\varepsilon}}+\frac{d_{TE_\mu}(z;\omega_k)}{\max d_{TE_\mu}}\right)$$	(Park, 2012, Park, 2012)
Structural/optimization (linear)	$$G(x_0)=2\nu_1\frac{(\nu_0-\nu_1)}{(\nu_0+\nu_1)}\pi(\nabla u_0(x_0)\cdot\nabla p_0(x_0))$$	Electric machine optimization (Gangl et al., 2013)

Topological derivatives constitute a rigorously justified, readily computable, and physically interpretable sensitivity concept. They enable non-iterative imaging and topology optimization across a spectrum of scientific and engineering applications, with their efficacy contingent on frequency, incident direction selection, normalization, and, for more complicated scenarios, on further algorithmic extension.