Presentation

Optimal sensor placement (OPS) in structural health monitoring (SHM)

Thyge Vinther Ludvigsen, s203591

15 Apr, 2026

Time plan

Big picture

  • What is SHM
  • What is vibration-based SHM
  • Problems in SHM

What is SHM

Structural Health Monitoring (SHM) - continuous assessment of structures using sensor data

Levels of SHM

  • Level 1: Damage vs no damage
  • Level 2: Localization of damage
  • Level 3: Quantification of damage

Methods of SHM

Global methods:

  • Vibration-based methods
  • Optical methods

Local methods:

  • Ultrasonic methods
  • …

Advantages of SHM

  • Real-time monitoring
  • Early detection of damage
  • Reduced maintenance costs
  • Improved safety
  • Reduced cost to same safety level

What is Vibration based SHM

General procedure

  • Gather sensor data
  • Operational Modal Analysis (OMA)
  • (Calculate Features)
  • Compare features with structural states

1

What is Vibration based SHM

Modal properties

  • Natural frequencies
  • Mode shapes
  • Damping ratios

What is Vibration based SHM

Advantages

  • Modal properties holds global information about the structure
  • Few sensors can be used to get estimate global information about the structure
  • Ambient excitation can be used to get the dynamic response of the structure.

Disadvantages

  • The stiffness is highly sensitive to environmental effects, which can lead to false positives.
  • Modal properties can be difficult to estimate accurately, especially in the presence of noise and environmental effects.
  • There is an large amount of data.

Problems in SHM

  • What is the Optimal Sensor Placement (OSP)?
  • Cost-Benefit Quantification?
  • Environmental effects?
  • Lack of SHM in practice?

My research focus

Optimal sensor placement (OPS) in structural health monitoring (SHM) whit the use of Bayesian risk.

Risk

\[ R(A) = P(A) \cdot L(A) \]

where:

  • \(P(A)\) is the probability of an event A occurring.
  • \(L(A)\) is the utility or loss do to the consequence associated with event A.

Bayes Risk

\[ r(d(x)) = E_{\theta}[E_{X|\theta}[L(\theta, d(x)) | \theta] \cdot f(\theta)] = \int \int L(\theta, d(x)) f(x|\theta) \space dx \space f(\theta) \space d\theta \]

where:

  • \(r(d(x))\) is the Bayes Risk. 1
  • \(d(x)\) is the decision rule that maps the data to a decision.
  • \(L(\theta, d(x))\) is the loss function that quantifies the loss incurred by making a decision \(d(x)\)
  • \(f(x|\theta)\) is the likelihood function that describes the probability of observing the data \(x\) given the parameter \(\theta\).

Why use Bayes Risk for OSP?

Problems in SHM

  • why use Bayes Risk for OSP?
    1. The Bayes framework Is perfect for Updating beliefs about the structural state based on new information.
    2. The Bayes risk is just expected risk for an given sensor configuration.
    3. The Bayes risk can be compared whit not using SHM, which can be used to quantify the benefit of using SHM.

Methodology

Big picture - how

Main Assumptions

  1. How to include stochastic in the model?
    • Sensor noise, Environmental effects, model uncertainty, etc.
  2. Set up an noise to feature model for an sensor configuration.
    • …

Big picture - how

Main workflow

  1. Set up the FEM model of the structure.
  2. Define damage states and sensor configurations.
  3. Define the likelihood function, priors, decision rul, and loss function.
  4. Estimate the modal properties for each damage state.
  5. Evaluate the Bayes risk for an given sensor configuration. whit the use of MCS sampling or numerical integration.
  6. Use an optimization algorithm to find the optimal sensor configuration that minimizes the Bayes risk.

Bayes Rule

\[ \text{Posterior} = \frac{\text{Likelihood} \cdot \text{Prior}}{\text{Evidence}} \Rightarrow f(\theta|x) = \frac{f(x|\theta)f(\theta)}{f(x)} \]

where:

  • \(f(\theta|x)\) is the posterior distribution of the parameter \(\theta\) given the data \(x\).
  • \(f(x|\theta)\) is the likelihood function, which describes the probability of observing the data \(x\) given the parameter \(\theta\).
  • \(f(\theta)\) is the prior distribution of the parameter \(\theta\), which represents our beliefs about the parameter before observing the data.
  • \(f(x)\) is the evidence, which can be found by the marginalization of the likelihood and prior.

Definition

Note: add some illustrations

  • Structural states: \(\theta \in \theta_0 ,..., \theta_k\)
  • No damage: \(H_0 \in \theta_0,..., \theta_n\) (Often: \(H_0 = \theta_0\))
  • Damage: \(H_1 \in \theta_{n+1},..., \theta_k\)
  • Sensor data: \(x(e)\)

Feature Model

Feature model: \[ y = g(x(e,\theta)) + \epsilon(e) \]

Uncertainty model: \[ \epsilon(e) \sim \mathcal{N}(0, \Sigma), \text{ Multivariate Gaussian noise} \]

where:

  • \(\Sigma\) is the covariance matrix of the noise
Symbol Meaning
\(H\) No damage \(H_0\), Damage \(H_1\)
\(e\) Sensor configuration
\(\theta\) Structural state
\(x(e)\) Sensor data
\(y = g(\theta, x(e, \theta)) + \epsilon(e)\) Feature vector
\(g(x(e, \theta))\) Features Function
\(\epsilon(e) \sim \mathcal{N}(0, \Sigma)\) Uncertainty
\(P(\theta)\), \(P(H)\) Prior probability
\(f(y \mid \theta)\), \(f(y \mid H)\) Likelihood of measurement
\(P(\theta \mid y)\), \(P(H \mid y)\) Posterior probability
\(L(d(...), H)\) Loss/Utility function
\(C(d(...), H)\) Cost function
\(d(...)\) Decision rule
\(\Psi(e)\) Bayes risk

Likelihood Function

Likelihood Function (Level 2-3 SHM system):

\[ f(y \mid \theta)= \frac{1}{\sqrt{(2\pi)^n |\Sigma|}} \exp\left( -\frac{1}{2} (y - g(\theta, e))^T \Sigma^{-1} (y - g(\theta, e)) \right) \]

Likelihood for the Level 1 SHM system:

\[ f(y \mid H_i)= \sum_{\theta \in H_i} f(y \mid \theta) P(\theta \mid H_i) \]

Symbol Meaning
\(H\) No damage \(H_0\), Damage \(H_1\)
\(e\) Sensor configuration
\(\theta\) Structural state
\(x(e)\) Sensor data
\(y = g(\theta, x(e, \theta)) + \epsilon(e)\) Feature vector
\(g(x(e, \theta))\) Features Function
\(\epsilon(e) \sim \mathcal{N}(0, \Sigma)\) Uncertainty
\(P(\theta)\), \(P(H)\) Prior probability
\(f(y \mid \theta)\), \(f(y \mid H)\) Likelihood of measurement
\(P(\theta \mid y)\), \(P(H \mid y)\) Posterior probability
\(L(d(...), H)\) Loss/Utility function
\(C(d(...), H)\) Cost function
\(d(...)\) Decision rule
\(\Psi(e)\) Bayes risk

Prior Distribution

Prior distribution (level 2-3 SHM system):

\[ P(\theta) \]

Prior distribution (level 1 SHM system):

\[ P(H) = \sum_{\theta \in H} P(\theta) \]

Symbol Meaning
\(H\) No damage \(H_0\), Damage \(H_1\)
\(e\) Sensor configuration
\(\theta\) Structural state
\(x(e)\) Sensor data
\(y = g(\theta, x(e, \theta)) + \epsilon(e)\) Feature vector
\(g(x(e, \theta))\) Features Function
\(\epsilon(e) \sim \mathcal{N}(0, \Sigma)\) Uncertainty
\(P(\theta)\), \(P(H)\) Prior probability
\(f(y \mid \theta)\), \(f(y \mid H)\) Likelihood of measurement
\(P(\theta \mid y)\), \(P(H \mid y)\) Posterior probability
\(L(d(...), H)\) Loss/Utility function
\(C(d(...), H)\) Cost function
\(d(...)\) Decision rule
\(\Psi(e)\) Bayes risk

Posterior Distribution

Posterior distribution (level 2-3 SHM system): \[ P(\theta \mid y) = \frac{f(y \mid \theta) P(\theta)}{f(y)} \]

Posterior distribution (level 1 SHM system):

\[ P(H_i \mid y) = \frac{f(y \mid H_i) P(H_i)}{f(y)} \]

Symbol Meaning
\(H\) No damage \(H_0\), Damage \(H_1\)
\(e\) Sensor configuration
\(\theta\) Structural state
\(x(e)\) Sensor data
\(y = g(\theta, x(e, \theta)) + \epsilon(e)\) Feature vector
\(g(x(e, \theta))\) Features Function
\(\epsilon(e) \sim \mathcal{N}(0, \Sigma)\) Uncertainty
\(P(\theta)\), \(P(H)\) Prior probability
\(f(y \mid \theta)\), \(f(y \mid H)\) Likelihood of measurement
\(P(\theta \mid y)\), \(P(H \mid y)\) Posterior probability
\(L(d(...), H)\) Loss/Utility function
\(C(d(...), H)\) Cost function
\(d(...)\) Decision rule
\(\Psi(e)\) Bayes risk

Decision Rule

Decision rule - Maximum a posteriori (MAP): \[ d(y) = \arg\max_{H_i} P(H_i \mid y) \]

Symbol Meaning
\(H\) No damage \(H_0\), Damage \(H_1\)
\(e\) Sensor configuration
\(\theta\) Structural state
\(x(e)\) Sensor data
\(y = g(\theta, x(e, \theta)) + \epsilon(e)\) Feature vector
\(g(x(e, \theta))\) Features Function
\(\epsilon(e) \sim \mathcal{N}(0, \Sigma)\) Uncertainty
\(P(\theta)\), \(P(H)\) Prior probability
\(f(y \mid \theta)\), \(f(y \mid H)\) Likelihood of measurement
\(P(\theta \mid y)\), \(P(H \mid y)\) Posterior probability
\(L(d(...), H)\) Loss/Utility function
\(C(d(...), H)\) Cost function
\(d(...)\) Decision rule
\(\Psi(e)\) Bayes risk

Loss Function

Utility / Loss function: \[ L(\theta, d(x)) = C(H_i, d(y)) \]

Cost Matrix: \[ C(H_i, d(y)) = \begin{bmatrix} C_{00} & C_{01} \\ C_{10} & C_{11} \end{bmatrix} \]

  • \(C_{00}\) is the cost of true negative (correctly identifying no damage)
  • \(C_{01}\) is the cost of false positive (incorrectly identifying damage)
  • \(C_{10}\) is the cost of false negative (incorrectly identifying no damage)
  • \(C_{11}\) is the cost of true positive (correctly identifying damage)
Symbol Meaning
\(H\) No damage \(H_0\), Damage \(H_1\)
\(e\) Sensor configuration
\(\theta\) Structural state
\(x(e)\) Sensor data
\(y = g(\theta, x(e, \theta)) + \epsilon(e)\) Feature vector
\(g(x(e, \theta))\) Features Function
\(\epsilon(e) \sim \mathcal{N}(0, \Sigma)\) Uncertainty
\(P(\theta)\), \(P(H)\) Prior probability
\(f(y \mid \theta)\), \(f(y \mid H)\) Likelihood of measurement
\(P(\theta \mid y)\), \(P(H \mid y)\) Posterior probability
\(L(d(...), H)\) Loss/Utility function
\(C(d(...), H)\) Cost function
\(d(...)\) Decision rule
\(\Psi(e)\) Bayes risk

Bayes Risk

Bayes Risk: \[ \Psi(e) = \mathbb{E}_{H, y}\left[ L(d(y), H_i) \right] = \sum_{H_i} P(H_i) \int f(y \mid H_i) L(d(y), H_i) dy \]

Optimal Sensor Configuration: \[ e^* = \arg\min_e \Psi(e) \]

Symbol Meaning
\(H\) No damage \(H_0\), Damage \(H_1\)
\(e\) Sensor configuration
\(\theta\) Structural state
\(x(e)\) Sensor data
\(y = g(\theta, x(e, \theta)) + \epsilon(e)\) Feature vector
\(g(x(e, \theta))\) Features Function
\(\epsilon(e) \sim \mathcal{N}(0, \Sigma)\) Uncertainty
\(P(\theta)\), \(P(H)\) Prior probability
\(f(y \mid \theta)\), \(f(y \mid H)\) Likelihood of measurement
\(P(\theta \mid y)\), \(P(H \mid y)\) Posterior probability
\(L(d(...), H)\) Loss/Utility function
\(C(d(...), H)\) Cost function
\(d(...)\) Decision rule
\(\Psi(e)\) Bayes risk

Summary

Level 2

flowchart TB
    subgraph SensorDesign["Sensor configurations: e    ."]
        direction TB
        style SensorDesign fill:#e8f5e9,stroke:#2e7d32,stroke-width:3px,color:#1b5e20
        
        subgraph StructuralState["Structural states: θ    ."]
            direction TB
            style StructuralState fill:#e3f2fd,stroke:#1565c0,stroke-width:3px,color:#0d47a1
            
            subgraph Measurement["Feature space: y    ."]
                direction TB
                style Measurement fill:#fff3e0,stroke:#e65100,stroke-width:3px,color:#bf360c
                
                A["Features<br/>y = g(θ, e) + ε(e)"]
                B["Likelihood<br/>f(y | θ)"]
                C["Posterior<br/>P(θ | y)"]
                D["Decision Rule<br/>d = max P(θ | y)"]
                E["Frequentist Risk<br/> R(d) = ∫ f(y | θ) L(θ, d) dy"]

                A --> B
                B --> C
                C --> D
                D --> E
            end

            F["Bayes Risk<br/> Ψ(e) = ∑<sub>θ</sub>  P(θ) R(d)"]
            E --> F
        end

        H["Optimization<br/>e* = argmin Ψ(e)"]
        F --> H
    end

Level 2 SHM system

Level 1

flowchart TB
    subgraph SensorDesign["Sensor configurations: e    ."]
        direction TB
        style SensorDesign fill:#e8f5e9,stroke:#2e7d32,stroke-width:3px,color:#1b5e20
        
        subgraph DamageNoDamage["Structural categories: H    ."]
            direction TB
            style DamageNoDamage fill:#fff9c4,stroke:#f57f17,stroke-width:3px,color:#e65100
            
            subgraph Measurement["Feature space: y    ."]
                direction TB
                style Measurement fill:#fff3e0,stroke:#e65100,stroke-width:3px,color:#bf360c
            

                subgraph StructuralState["Structural states: θ    ."]
                    direction TB
                    style StructuralState fill:#e3f2fd,stroke:#1565c0,stroke-width:3px,color:#0d47a1
                
                    A["Features<br/>y = g(θ, e) + ε(e)"]
                    B["Likelihood<br/>f(y | H) = ∑<sub>θ</sub> f(y | θ) P(θ | H)"]
                    A --> B
                    end

            C["Posterior<br/>P(H | y)"]
            D["Decision Rule<br/>d = max P(H | y)"]
            E["Frequentist Risk<br/> R(d) = ∫ f(y | H) L(H, d) dy"]

            B --> C
            C --> D
            D --> E
            end

            F["Bayes Risk<br/> Ψ(e) = ∑<sub>H</sub>  P(H) R(d)"]
            E --> F
        end

        H["Optimization<br/>e* = argmin Ψ(e)"]
        F --> H
    end

Level 1 SHM system

Most important

Uncertainty link whit sensor placement: \[ \epsilon(e) = ??? \]

Prior probability: \[ P(\theta) = ??? \]

Cost matrix: \[ C(\theta, d(x)) = \begin{bmatrix} C_{00} & C_{01} \\ C_{10} & C_{11} \end{bmatrix} \]

Symbol Meaning
\(H\) No damage \(H_0\), Damage \(H_1\)
\(e\) Sensor configuration
\(\theta\) Structural state
\(x(e)\) Sensor data
\(y = g(\theta, x(e, \theta)) + \epsilon(e)\) Feature vector
\(g(x(e, \theta))\) Features Function
\(\epsilon(e) \sim \mathcal{N}(0, \Sigma)\) Uncertainty
\(P(\theta)\), \(P(H)\) Prior probability
\(f(y \mid \theta)\), \(f(y \mid H)\) Likelihood of measurement
\(P(\theta \mid y)\), \(P(H \mid y)\) Posterior probability
\(L(d(...), H)\) Loss/Utility function
\(C(d(...), H)\) Cost function
\(d(...)\) Decision rule
\(\Psi(e)\) Bayes risk

Least Important / Assumptions

Decision Rule: (Maximum a posteriori (MAP)) \[ d(y) = \arg\max_{H_i} P(H_i \mid y) \]

Computation of the Bayes Risk:

  • MCS sampling
  • Numerical integration

Optimization algorithm: \[ ??? \]

Symbol Meaning
\(H\) No damage \(H_0\), Damage \(H_1\)
\(e\) Sensor configuration
\(\theta\) Structural state
\(x(e)\) Sensor data
\(y = g(\theta, x(e, \theta)) + \epsilon(e)\) Feature vector
\(g(x(e, \theta))\) Features Function
\(\epsilon(e) \sim \mathcal{N}(0, \Sigma)\) Uncertainty
\(P(\theta)\), \(P(H)\) Prior probability
\(f(y \mid \theta)\), \(f(y \mid H)\) Likelihood of measurement
\(P(\theta \mid y)\), \(P(H \mid y)\) Posterior probability
\(L(d(...), H)\) Loss/Utility function
\(C(d(...), H)\) Cost function
\(d(...)\) Decision rule
\(\Psi(e)\) Bayes risk

Any Questions?