[Paper Review] Dimension reduction in MHD power generation models: dimensional analysis and active subspaces

Andrew Glaws, Paul G. Constantine+2 more|arXiv (Cornell University)|Sep 5, 2016

Model Reduction and Neural Networks15 references3 citations

TL;DR

This paper proposes a combined approach using dimensional analysis and active subspaces to reduce input dimensionality in MHD power generation models, revealing that only four or fewer linear combinations of log-transformed parameters significantly affect key outputs. The method identifies dominant input sensitivities—such as viscosity and pressure gradient for flow velocity, and resistivity and magnetic field for induced magnetic field—enabling efficient uncertainty quantification and design optimization in computationally expensive simulations.

ABSTRACT

Magnetohydrodynamics (MHD)---the study of electrically conducting fluids---can be harnessed to produce efficient, low-emissions power generation. Today, computational modeling assists engineers in studying candidate designs for such generators. However, these models are computationally expensive, so studying the effects of the model's many input parameters on output predictions is typically infeasible. We study two approaches for reducing the input dimension of the models: (i) classical dimensional analysis based on the inputs' units and (ii) active subspaces, which reveal low-dimensional subspaces in the space of inputs that affect the outputs the most. We also review the mathematical connection between the two approaches that leads to consistent application. The dimension reduction yields insights into the driving factors in the MHD power generation models. We study both the simplified Hartmann problem, which admits closed form expressions for the quantities of interest, and a large-scale computational model with adjoint capabilities that enable the derivative computations needed to estimate the active subspaces.

Motivation & Objective

• To address the computational infeasibility of studying high-dimensional input spaces in MHD power generation simulations.
• To identify low-dimensional input subspaces that most influence quantities of interest such as average flow velocity and induced magnetic field.
• To establish a mathematical connection between classical dimensional analysis and gradient-based active subspaces for consistent dimension reduction.
• To apply the framework to both analytical (Hartmann problem) and large-scale 3D computational MHD models to validate insights.
• To enable efficient uncertainty quantification and computational design by revealing dominant input sensitivities and intrinsic model dimensionality.

Proposed method

• Apply classical dimensional analysis to the MHD governing equations to determine the maximum number of dimensionless groups (unitless parameters) affecting the system.
• Use adjoint-based sensitivity analysis to compute gradients of outputs with respect to inputs in a large-scale 3D MHD simulation with full derivative capabilities.
• Construct the active subspace using the eigendecomposition of the matrix $\mathbf{C} = \mathbb{E}[\nabla f(\mathbf{x}) \nabla f(\mathbf{x})^T]$, where $f(\mathbf{x})$ is the quantity of interest.
• Log-transform the input parameters to linearize the relationship between inputs and outputs, enabling comparison with dimensional analysis results.
• Rank input combinations by their influence on output variance using eigenvalues of the active subspace matrix.
• Visualize the low-dimensional structure via one- and two-dimensional summary plots of the first active variable(s) against the output.

Experimental results

Research questions

• RQ1 How many effective input parameters govern the average flow velocity and induced magnetic field in MHD power generation models?
• RQ2 What is the mathematical relationship between dimensional analysis and active subspaces in physical systems with multiple input parameters?
• RQ3 Which input parameters most strongly influence the output quantities of interest in both simplified and large-scale MHD models?
• RQ4 To what extent can the output behavior be approximated by a one-dimensional function of a single linear combination of log-transformed inputs?
• RQ5 In which regions of the parameter space does the one-dimensional approximation break down, requiring higher-dimensional representation?

Key findings

• The Hartmann problem exhibits an intrinsic dimension of 2 for both average flow velocity and induced magnetic field, meaning only two linear combinations of log-transformed inputs govern these outputs.
• The large-scale 3D MHD model has an intrinsic dimension of at most 4, as bounded by dimensional analysis, with active subspaces confirming this upper limit.
• The first eigenvector of the active subspace matrix reveals that average flow velocity is most sensitive to fluid viscosity and applied pressure gradient.
• The induced magnetic field is most sensitive to resistivity and applied magnetic field, with fluid density having negligible influence.
• In regions of low viscosity, high pressure gradient, low resistivity, and high magnetic field, the induced magnetic field requires more than two linear combinations for accurate approximation.
• One-dimensional summary plots show that both average flow velocity and induced magnetic field in the Hartmann problem can be well approximated by a single linear combination of log-transformed inputs, indicating strong low-dimensional structure.