2. CAE simulation fundamentals for surrogate modeling

A surrogate model is only as trustworthy as the simulation data it learns from. Before choosing a regression architecture or designing a sampling plan, an engineer must understand what the CAE solver actually computes, how the continuous physics is reduced to a finite numerical problem, and which scalar or field quantities will serve as prediction targets. This chapter establishes that foundation: the governing equations behind common CAE disciplines, the discretization pipeline that turns them into algebraic systems, and the mapping from raw solver output to well-defined quantities of interest (QoIs) suitable for surrogate training.

Governing Equations: The Physics the Solver Approximates

Most structural, thermal, and flow simulations rest on a small set of conservation laws. For solid mechanics, the steady-state equilibrium statement is the balance of internal stress divergence and body force, ∇·σ + b = 0, closed by a constitutive law (for linear elasticity, σ = C : ε) and the strain–displacement relation ε = ½(∇u + ∇uᵀ). For heat transfer, the steady conduction equation is ∇·(k∇T) + q = 0. For incompressible fluid flow, the Navier–Stokes system couples momentum conservation with mass conservation, ∇·u = 0. Each system is completed by boundary conditions: prescribed displacement or traction, fixed temperature or heat flux, inlet velocity or outlet pressure.

Two implications matter for surrogate modeling. First, the QoI is a functional of the solution field, not the field itself — a maximum von Mises stress, a peak temperature, a drag coefficient. Second, the input space is defined by the parameters that enter these equations and their boundary conditions: geometry dimensions, material properties such as Young's modulus E and Poisson ratio ν, load magnitudes, and boundary condition values. A surrogate maps a vector of such parameters to one or more QoIs. If a parameter does not appear in the governing system or the boundary conditions, it cannot influence the QoI, and including it only adds noise to the training set.

Discretization: From PDE to Algebraic System

Analytical solutions exist for a handful of canonical geometries; everything else requires discretization. The finite element method (FEM) partitions the domain into elements, approximates the field within each element by shape functions, and enforces the governing equation in a weighted residual sense. The result is a linear system K u = f for linear problems, or a nonlinear residual R(u) = 0 solved by Newton iteration for nonlinear ones. Finite volume methods (FVM), standard in CFD, integrate the conservation law over each control volume and enforce flux balance across faces, producing a system that is naturally conservative.

The practical consequence is that every QoI carries discretization error. Mesh refinement reduces it, but at a cost that scales roughly with the number of degrees of freedom. For surrogate datasets, this creates a design decision: use a consistent mesh across all samples so that discretization error is a fixed bias rather than sample-dependent noise, or use adaptive meshing and accept that part of the QoI variation reflects mesh changes rather than physics. The first option is almost always preferable for regression, because a surrogate trained on mixed-fidelity data learns artifacts of the meshing strategy.

A useful sanity check before generating any dataset: run two meshes of the same design (e.g., a coarse and a refined mesh) and compare the QoI. If the difference exceeds a few percent of the expected QoI range across the design space, the mesh is too coarse and the surrogate will learn mesh error rather than physics.

Output Quantities of Interest: From Fields to Scalars

Solvers produce nodal or cell-centered fields — displacement vectors, stress tensors, temperature, pressure, velocity. Surrogates, however, are typically trained on a modest number of scalars. The extraction step is therefore part of the modeling problem, not an afterthought. Common QoI categories include:

  • Extremal values: maximum von Mises stress, peak temperature, minimum safety factor. These are non-smooth functionals of the field — the location of the maximum can jump between designs — which makes them harder to regress than volume integrals.
  • Integral quantities: total strain energy, heat flow through a boundary, drag and lift coefficients. These are smoother in the design parameters and generally easier for a surrogate to learn.
  • Point or region probes: displacement at a monitored node, average pressure over an outlet patch. Sensitive to probe placement; define the probe location in a way that is invariant across geometry variations.
  • Field outputs: full pressure or stress distributions, handled by dimensionality reduction (e.g., POD/PCA) before regression. These require a common mesh or a mapping step.

The smoothness of the parameter-to-QoI map directly affects surrogate accuracy. A maximum-stress QoI may exhibit kinks when the critical location switches from one fillet to another; a strain-energy QoI will not. Where possible, include both a smooth integral measure and the engineering-critical extremal measure, and inspect the response surface of each before committing to a model class.

CAE simulation pipeline from governing equations to surrogate-ready quantities of interest A left-to-right diagram showing design parameters entering the governing equations, discretization into an algebraic system, solver output fields, and QoI extraction into scalars for surrogate training, with a feedback loop for mesh consistency. Design Parameters geometry dims E, ν, k loads, BC values x ∈ ℝⁿ Governing Equations ∇·σ + b = 0 ∇·(k∇T) + q = 0 ∇·u = 0 + boundary conditions Discretization FEM / FVM mesh K u = f R(u) = 0 (Newton) consistent mesh → fixed bias Solution Fields u(x), σ(x) T(x), p(x), v(x) nodal / cell-centered QoI Extraction → Scalar Targets • Extremal: max σ_vm, T_peak, min SF • Integral: strain energy, heat flow, C_d, C_l • Probe: u at node, avg p over patch • Field: POD/PCA coefficients smoothness of x → QoI governs model choice Surrogate Model f̂ : x → QoI training set {(xᵢ, QoIᵢ)} validation vs. held-out CAE runs mesh refinement check: ΔQoI across mesh levels physics numerics solver output QoI

Worked Example: A Cantilever Bracket Dataset

Consider a steel cantilever bracket with a fillet radius r, a web thickness t, and a tip load F. The design vector is x = (r, t, F) with ranges r ∈ [4, 12] mm, t ∈ [3, 8] mm, F ∈ [500, 2000] N; material is structural steel with E = 210 GPa, ν = 0.3. The QoIs are maximum von Mises stress σ_max and tip deflection δ_tip. A reproducible dataset construction proceeds as follows.

  1. Mesh convergence study. Fix a mid-range design (r = 8 mm, t = 5 mm, F = 1200 N) and run three mesh levels, e.g., global element size 4 mm, 2.5 mm, and 1.5 mm with local refinement at the fillet. Record σ_max and δ_tip for each. Accept the finest mesh when the change from the previous level is below 2% for both QoIs; use that mesh template for all subsequent samples.
  2. Parameterized geometry. Drive the CAD model from r and t so that every sample regenerates the same topology. Verify that the mesher produces a comparable element count across the design space; large swings in element count indicate that the mesh template is not robust to geometry changes.
  3. Sampling. Draw a space-filling design (e.g., Latin hypercube) of 60–100 points in the three-dimensional parameter space. For each point, run the solver, extract σ_max and δ_tip, and store the parameter vector alongside the QoIs and a solver status flag.
  4. Quality screening. Reject samples with non-converged Newton iterations, negative Jacobians, or element distortion warnings. A surrogate trained on failed or marginal solves will inherit those artifacts.
  5. Response inspection. Plot σ_max against each parameter. A smooth monotonic trend in t and F is expected; a kink in r may indicate that the critical stress location migrates from the fillet to the fixed root, which is physically meaningful and should be preserved rather than smoothed away.
Do not mix solvers, element formulations, or mesh templates within one training set unless fidelity is an explicit input variable. A surrogate cannot distinguish a genuine physical effect from a change in discretization strategy, and the resulting model will fail unpredictably on new designs.

What the Surrogate Actually Learns

The mapping from x to a QoI is a composition: parameter vector → geometry and boundary conditions → discretized system → solution field → extracted scalar. Each arrow introduces its own error and non-smoothness. Geometry regeneration can fail or produce slightly different topology; the solver introduces discretization and iteration error; extraction of an extremal QoI introduces location-switching discontinuities. A well-constructed surrogate dataset controls the first two sources (consistent template, convergence-checked mesh) and characterizes the third (response-surface inspection) before any regression model is fitted. With that foundation in place, the choice of surrogate architecture becomes a question of approximating a known, reasonably smooth function rather than compensating for uncontrolled noise in the training data.

← Previous 📖 Contents Next →