12. Verification, Validation and Uncertainty Quantification

Introduction and Context

Computational fluid dynamics (CFD) produces numerical approximations to the governing equations of fluid motion. A colourful contour plot, a converged residual history, or a close agreement with an experimental curve does not, by itself, establish that a simulation is credible. Credibility requires a disciplined distinction between whether the equations have been solved correctly, whether the mathematical model represents reality adequately, and how uncertain the reported prediction is.

These concerns are addressed through verification, validation and uncertainty quantification (VVUQ). Verification asks, “Are the equations being solved correctly?” Validation asks, “Are the correct equations being solved for the intended physical system?” Uncertainty quantification asks, “How much confidence should be placed in the prediction, and how do uncertain inputs affect it?”

The distinction is especially important in postgraduate engineering practice. A simulation may be numerically accurate but physically inappropriate, or physically well modelled but poorly resolved on the computational mesh. Design decisions involving safety, efficiency, emissions, thermal limits or structural loading must therefore be supported by evidence rather than by visual plausibility.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Distinguish code verification, solution verification and validation.

2. Design benchmark and manufactured-solution tests for CFD codes.

3. Assess iterative, spatial and temporal discretisation errors.

4. Apply Richardson extrapolation and the grid-convergence index (GCI).

5. Select meaningful validation metrics and compare simulations with experiments.

6. Identify sources of uncertainty and propagate them through a CFD model.

7. Perform sensitivity analysis and interpret validation evidence responsibly.

8. Communicate assumptions, limitations and credibility claims transparently.

The Three Questions of Computational Credibility

Verification concerns the numerical implementation. Code verification examines whether the computer program correctly implements the intended algorithms and equations. Typical targets include flux calculations, boundary conditions, source terms, turbulence models, time integration and linear solvers. Code verification is normally performed using problems with known analytical or manufactured solutions.

Solution verification concerns the numerical solution generated for a particular case. It evaluates errors caused by finite mesh size, finite time step, incomplete iterative convergence and, in some cases, round-off or solver tolerances. Solution verification does not determine whether the physical model is appropriate; it determines whether the implemented model has been solved sufficiently accurately.

Validation concerns the relationship between computation and the physical world. It involves comparing predictions with reliable experimental or field data under comparable conditions. Validation is conditional: a model may be adequate for one flow regime, geometry and output quantity but inadequate for another. Validation is therefore not a universal certification of a code or model.

A useful hierarchy is:

• Mathematical model: conservation laws, constitutive relations and physical closures.
• Numerical method: discretisation, algorithms and implementation.
• Computed solution: the result for a specified mesh, time step and convergence tolerance.
• Physical observation: measured data with its own uncertainty.
• Engineering decision: a conclusion based on both prediction and uncertainty.

Confusing these levels is a common source of overconfidence. Agreement with an experiment can result from compensating errors, while disagreement may arise from experimental uncertainty, geometric differences or inappropriate boundary conditions rather than from a coding error.

Verification of CFD Codes

Analytical and benchmark solutions

A code should first be tested against problems whose solution is known or independently established. Examples include fully developed laminar flow in a plane channel or circular pipe, Couette flow, natural convection in a differentially heated cavity, inviscid potential flow and one-dimensional shock-tube problems. These tests isolate particular physical and numerical features.

A benchmark is useful only when its specification is precise. The geometry, fluid properties, initial and boundary conditions, nondimensional parameters, reference solution and comparison quantities should be documented. For example, a channel-flow test may compare the computed velocity profile, pressure gradient and wall shear stress, rather than merely checking that the flow “looks parabolic.”

Method of manufactured solutions

The method of manufactured solutions (MMS) is a general code-verification technique. An analytical field is selected deliberately, even if it is not a natural solution of the unforced equations. Substitution of the selected field into the governing equations generates source terms. The code is then run with these source terms, and the numerical solution is compared with the manufactured field.

For a scalar transport equation,

∂φ/∂t + ∇·(uφ) = ∇·(Γ∇φ) + S,

one may prescribe φ_m(x,y,t), calculate the source S_m required to make it an exact solution, and impose boundary and initial data derived from φ_m. If the code is correct and the discretisation is of order p, the error should decrease approximately as h^p when the characteristic mesh spacing h is reduced in the asymptotic range.

MMS can expose errors that ordinary benchmark tests fail to reveal, including incorrect source-term integration, sign errors, inconsistent boundary conditions and implementation mistakes in multidimensional operators. It should be applied progressively: first to individual operators, then to coupled equations, and finally to representative multiphysics configurations.

Verification results are strengthened by reporting error norms. For an error e_i at discrete locations, common measures are

L₁ = (1/N) Σ|e_i|, L₂ = [(1/N)Σe_i²]¹ᐟ², L∞ = max|e_i|.

The norm must be defined clearly because different norms emphasize different features. L∞ is sensitive to local peaks, whereas L₂ represents an average global error.

Code-verification evidence

A mature verification programme includes automated regression tests, unit tests for individual discretisation components, conservation checks and documented benchmark comparisons. Regression tests are particularly valuable after modifications to parallelisation, compiler settings, turbulence closures or mesh-handling routines. Passing a regression test means that behaviour has remained consistent; it does not necessarily mean that the behaviour is physically correct.

Solution Verification: Establishing Numerical Adequacy

A computed result contains several numerical error components. A useful conceptual decomposition is

E_total ≈ E_iterative + E_spatial + E_temporal + E_modelling + E_input,

where iterative, spatial and temporal terms belong mainly to solution verification, while modelling and input errors are addressed through validation and uncertainty analysis. The terms are not always independent, and their interactions may be significant.

Iterative convergence

Iterative convergence concerns the reduction of algebraic residuals and changes in monitored quantities. Residuals measure the imbalance of discretised equations, but a small residual does not automatically imply a small error in a quantity of interest. A drag coefficient may have stabilised while a local recirculation structure continues to change.

Convergence assessment should therefore use several indicators:

• scaled residual histories;
• global mass and energy imbalance;
• stability of relevant integral outputs;
• changes in local values at critical locations; and
• sensitivity to solver tolerances or under-relaxation.

For unsteady flows, periodic statistical convergence may be more appropriate than convergence to a steady state. Mean values, root-mean-square fluctuations and spectra should be evaluated over a sufficiently long sampling interval after transients have decayed.

Spatial discretisation and grid studies

A grid study examines how a quantity of interest changes as the mesh is refined. At least three systematically refined meshes are preferable for estimating an observed order of convergence. The refinement ratio is commonly defined as

r = h_coarse / h_fine > 1.

For a quantity φ, the observed order from three solutions can be estimated by comparing successive differences, subject to the assumptions of monotonic convergence and an asymptotic refinement range. In practice, mesh refinement should preserve important geometric and physical features: boundary-layer resolution, wake structure, curvature, shock location and local mesh quality.

A mesh-independent-looking result can be misleading. Local errors may remain large even when a global force changes little, and cancellation of errors can create apparent agreement between two meshes. Grid studies should therefore be performed for each major quantity of interest, not only for residuals or visual contours.

Richardson extrapolation and the GCI

If the solution is in the asymptotic range and the leading discretisation error behaves as h^p, Richardson extrapolation estimates the zero-grid-spacing value:

φ_ext ≈ φ_f + (φ_f − φ_c)/(r^p − 1),

where φ_f and φ_c are fine- and coarse-grid results. The approximate relative error between fine and coarse solutions is

e_a = |(φ_f − φ_c)/φ_f|.

The grid-convergence index expresses a conservative estimate of discretisation uncertainty:

GCI_fine = F_s e_a/(r^p − 1),

where F_s is a safety factor, often selected according to the number of grids and the quality of the refinement study. The GCI should be reported as an uncertainty band or percentage associated with a specified quantity, not as a universal property of the mesh.

For example, suppose a pressure-drop prediction is 102 Pa on a coarse mesh and 100 Pa on a fine mesh, with r = 2 and p = 2. The approximate relative difference is 2%, and the estimated fine-grid discretisation uncertainty is approximately F_s(0.02)/(4−1). With F_s = 1.25, this is about 0.83%. This calculation is meaningful only if the meshes are systematically related and the solutions are in the asymptotic range.

Non-monotonic results require caution. If successive values oscillate, the assumptions behind simple Richardson extrapolation may not hold. Possible causes include changing separation topology, inadequate refinement near a discontinuity, mesh-induced alignment effects or insufficient iterative convergence.

Temporal discretisation

In transient CFD, reducing the time step is analogous to refining the spatial grid. A time-step study should monitor the output of interest while maintaining adequate spatial resolution. The Courant number provides a useful stability and resolution indicator, but satisfying a stability limit does not guarantee temporal accuracy. Turbulent fluctuations, vortex shedding and acoustic waves may require time steps much smaller than a nominal stability requirement.

A practical study varies Δt systematically and compares phase, amplitude, mean values and frequency content. For periodic flows, a simulation that predicts the correct mean force but an incorrect shedding frequency is not fully verified for unsteady prediction.

Validation Against Physical Evidence

Validation compares a prediction with observations for a defined purpose. The comparison must begin with validation quantities and validation conditions. A simulation of an airfoil, for example, may be validated for lift and moment over a specified Reynolds-number and angle-of-attack range; that evidence should not automatically be extended to stall dynamics or surface heat transfer.

Quality of experimental data

Experimental measurements have bias, random variability, calibration uncertainty, spatial resolution limits and possible disturbance from instrumentation. The computational geometry and operating conditions should match the experiment, including blockage, surface roughness, inlet turbulence, temperature, pressure and transition state where relevant. Small differences in these inputs can dominate the apparent CFD error.

Validation is strongest when the data set includes multiple operating conditions and when the experiment is independent of the data used to tune model parameters. Calibration and validation should not be conflated. If constants are adjusted using the same measurements used to assess performance, the resulting agreement is not an independent validation.

Validation metrics

The metric should reflect the engineering decision. Possible measures include absolute error, relative error, bias, root-mean-square error, mean absolute error, normalized error and confidence intervals. For a set of n observations y_i and predictions ŷ_i, the RMSE is

RMSE = [ (1/n)Σ(ŷ_i − y_i)² ]¹ᐟ².

For spatial profiles, pointwise differences, integrated norms and comparisons of important features such as peak position or boundary-layer thickness may all be relevant. For unsteady signals, phase error and spectral content should be considered. A single aggregate metric can conceal a serious local defect.

A validation comparison should show both computational and experimental uncertainty. If the prediction interval and measurement interval overlap, the result may be consistent with the experiment; this does not prove exactness. Conversely, a statistically significant difference may still be acceptable for a design if it is smaller than the engineering tolerance, while a small average error may be unacceptable at a safety-critical location.

Uncertainty Quantification and Sensitivity Analysis

Uncertainty enters a CFD analysis through geometry, material properties, boundary and initial conditions, numerical settings, model-form assumptions and experimental reference data. Aleatory uncertainty describes inherent variability, such as fluctuating inlet turbulence. Epistemic uncertainty arises from limited knowledge, such as an uncertain transition criterion or poorly known roughness.

Propagation methods

For a response Y = f(X₁, X₂, …, X_k), local linear propagation approximates the variance as

σ_Y² ≈ ΣΣ (∂f/∂X_i)(∂f/∂X_j) Cov(X_i,X_j).

This method is efficient but assumes sufficiently smooth behaviour and relatively small uncertainties. Monte Carlo simulation samples uncertain inputs from prescribed probability distributions, runs the model repeatedly and estimates the output distribution. It is conceptually simple but can be expensive for high-fidelity CFD.

Latin hypercube sampling improves coverage of the input space with fewer samples than unstructured random sampling. Polynomial chaos, Gaussian-process surrogates and reduced-order models can further reduce computational cost. Surrogate models must themselves be verified over the design space, especially near discontinuities or regime changes.

Sensitivity analysis

Sensitivity analysis identifies which inputs most influence a response. Local derivatives are useful near a nominal operating point; global methods, such as variance-based indices, assess influence across the full uncertain input range and can reveal interaction effects. Sensitivity results guide mesh refinement, experimental design and model-development priorities.

A parameter with high nominal sensitivity is not necessarily the dominant uncertainty source: its input distribution may be narrow. Conversely, a moderately sensitive parameter with large uncertainty may dominate the output variance. Decisions should therefore consider both sensitivity and uncertainty magnitude.

Combining numerical and physical uncertainties

Numerical uncertainty should be reduced until it is small relative to modelling and input uncertainty, unless computational cost makes this impractical. A common engineering approximation combines independent standard uncertainties by root-sum-square:

U_total ≈ (U₁² + U₂² + … + U_m²)¹ᐟ².

This expression must not be used mechanically when uncertainties are correlated, non-Gaussian or strongly nonlinear. Model-form uncertainty is particularly difficult to quantify and may require comparison of alternative turbulence, transition or multiphase models rather than a single statistical error bar.

An Integrated VVUQ Workflow

A defensible CFD study can follow the sequence below, with iteration where evidence reveals deficiencies:

1. Define the decision, quantities of interest and acceptable error.

2. Specify the physical model, assumptions, parameter ranges and boundary conditions.

3. Verify relevant code components using analytical solutions, benchmarks and MMS.

4. Demonstrate iterative convergence and conservation for the actual case.

5. Conduct mesh and time-step studies and estimate discretisation uncertainty.

6. Compare with independent experimental or trusted reference data.

7. Quantify input, numerical and, where possible, model-form uncertainty.

8. Perform sensitivity analysis to identify influential assumptions.

9. Report predictions, uncertainty intervals, validation metrics and limitations.

10. Repeat the assessment when the geometry, regime or intended use changes materially.

This workflow is not a bureaucratic sequence. It prevents effort from being spent on high-fidelity simulations whose basic implementation or numerical adequacy has not been established.

Illustrative Case Study: CFD Assessment of a Compact Heat Exchanger

Background

Consider an illustrative industrial study of a compact air-to-liquid heat exchanger. The engineering objective is to predict pressure drop and heat-transfer rate at three air flow rates so that a fan and thermal control system can be selected.

Situation and problem

An initial steady RANS simulation used a coarse mesh and reported a pressure drop within 3% of a manufacturer curve. The result appeared satisfactory, but the manufacturer’s curve did not specify uncertainty, and the computational model used an idealised inlet and smooth walls. The engineering team needed to determine whether the apparent agreement was credible.

Analysis and approach

First, the team defined pressure drop, total heat-transfer rate and outlet temperature as quantities of interest. A laminar channel benchmark and an MMS test for the energy equation were used to check diffusion, convection and source-term implementation. The actual case was then solved on three systematically refined meshes, with near-wall resolution controlled consistently. Residuals were reduced until pressure drop and heat transfer changed negligibly under further iterations.

The fine-to-medium differences were smaller than the medium-to-coarse differences, suggesting an asymptotic trend for pressure drop. A GCI was calculated separately for pressure drop and heat transfer; the latter showed a larger discretisation uncertainty because temperature gradients near the fins were less well resolved. A time-step study was unnecessary for the final steady model but was performed on a transient start-up calculation used to assess thermal response.

Validation data were obtained from a controlled test at matching flow rates, inlet temperature and fluid properties. Measurement uncertainty was estimated from sensor calibration and flow-meter specifications. The comparison showed that pressure-drop predictions lay within the combined computational and measurement uncertainty, whereas heat-transfer predictions were consistently high at the largest flow rate.

Sensitivity analysis identified inlet turbulence intensity and fin-surface roughness as influential parameters. Additional simulations using plausible ranges reduced the heat-transfer bias but increased the predicted uncertainty interval. The team concluded that the model was adequate for preliminary fan sizing, but not yet sufficiently reliable for guaranteeing the thermal duty at the highest flow rate.

Outcome and learning points

The original 3% agreement was not accepted as validation by itself. The final report distinguished numerical uncertainty from uncertainty in roughness, inlet conditions and turbulence modelling. It recommended a conservative design margin and further experiments under high-flow conditions.

The case illustrates four important principles: agreement with one reference curve is not sufficient; each output requires its own verification evidence; uncertainty may be dominated by physical assumptions rather than mesh resolution; and the credibility claim must be matched to the intended engineering decision.

Common Errors, Limitations and Professional Responsibilities

Frequent mistakes include declaring grid independence after only one mesh comparison, using residual reduction as the sole convergence criterion, changing several numerical settings simultaneously, comparing unlike experimental conditions, tuning a model against the validation data, and reporting excessive significant figures. Another mistake is treating a turbulence model as an experimentally verified physical law across all flow regimes.

Verification and validation also have practical limits. High-Reynolds-number industrial flows may lack sufficiently resolved experimental data. Model-form uncertainty can be larger than discretisation uncertainty and difficult to express probabilistically. Computational expense may restrict the number of samples in a global uncertainty analysis. These limitations should be stated explicitly rather than hidden behind a narrow error estimate.

Professional reporting should preserve reproducibility. A report should identify software version, solver settings, mesh statistics, wall treatment, convergence criteria, physical properties, boundary conditions, time-step information, validation data sources and uncertainty assumptions. Safety-critical applications may require independent review, configuration control, traceability and compliance with sector-specific standards. Engineers have an ethical responsibility not to present a numerically precise result as physically certain when evidence does not support that claim.

Digital Tools and Emerging Directions

Modern VVUQ increasingly uses automated mesh-convergence pipelines, continuous-integration regression tests, containerised environments and workflow managers. Statistical post-processing can generate uncertainty intervals and sensitivity rankings, while machine-learning surrogates reduce the cost of repeated simulations. However, automation does not remove the need for judgement: a surrogate trained on a narrow parameter range may fail near separation, transition or shock formation.

Current research is moving toward multifidelity uncertainty quantification, Bayesian calibration of uncertain model parameters, data assimilation, adaptive sampling and physics-informed surrogate models. These methods are valuable when experimental data are sparse, but they introduce additional assumptions about prior distributions, likelihood functions and model discrepancy. Their credibility depends on transparent validation just as conventional CFD does.

Best Practices for Credible CFD

A high-quality VVUQ study should:

• define quantities of interest before examining results;
• verify code features relevant to the application;
• separate iterative, discretisation, modelling and input uncertainties;
• use at least three systematically related meshes when estimating convergence trends;
• report numerical uncertainty for each important output;
• compare simulations with independent, well-characterised measurements;
• match geometry and operating conditions carefully;
• test sensitivity to influential assumptions;
• avoid false precision and disclose non-convergence or missing data; and
• connect the final credibility statement to a specific engineering decision.

Chapter Summary

Verification establishes whether the equations and algorithms have been implemented and solved correctly. Code verification uses analytical solutions, benchmarks and manufactured solutions; solution verification examines iterative, spatial and temporal discretisation errors. Grid studies, Richardson extrapolation and the GCI provide structured estimates of mesh-related uncertainty, provided their assumptions are satisfied. Validation compares predictions with independent physical evidence under matched conditions, using metrics appropriate to the quantity and decision. Uncertainty quantification accounts for uncertain inputs, numerical error and model-form limitations, while sensitivity analysis identifies the assumptions that most influence a result. Credible CFD is therefore not a single agreement value but a documented chain of evidence connecting implementation, numerical resolution, physical adequacy and engineering purpose.

Key Terms

Code verification: Demonstration that software correctly implements the intended mathematical equations and algorithms.

Solution verification: Assessment of numerical errors in a computed solution for a specified case.

Validation: Evaluation of how accurately a model represents a real physical system for a defined purpose.

Manufactured solution: An imposed analytical field used to generate source terms for code testing.

Benchmark problem: A standard problem with an accepted analytical or reference solution.

Quantity of interest: A reported output, such as drag, heat transfer or pressure drop, used for a decision.

Iterative convergence: Reduction of algebraic and nonlinear solution errors through solver iterations.

Discretisation error: Error caused by finite spatial or temporal resolution.

Richardson extrapolation: Estimation of the zero-grid-spacing solution from systematically refined results.

Grid-convergence index: A conservative estimate of grid-related numerical uncertainty.

Asymptotic range: Refinement range in which the leading discretisation error follows the expected order.

Validation metric: Quantitative measure used to compare prediction and observation.

Aleatory uncertainty: Variability inherent in a system or process.

Epistemic uncertainty: Uncertainty caused by incomplete knowledge or limited information.

Sensitivity analysis: Study of how changes in inputs affect a model output.

Model-form uncertainty: Uncertainty arising from approximations or inadequacy in the physical model.

Uncertainty propagation: Transfer of input uncertainties through a computational model to its outputs.

Surrogate model: An inexpensive approximation used to represent an expensive computational model.

Review Questions

Short Answer Questions

1. Distinguish code verification from solution verification.

2. What is the purpose of the method of manufactured solutions?

3. Why can a small residual fail to demonstrate convergence of a quantity of interest?

4. State two conditions required for meaningful use of Richardson extrapolation.

5. Differentiate aleatory and epistemic uncertainty.

Long Answer Questions

1. Explain the relationship among verification, validation and uncertainty quantification in a CFD investigation.

2. Describe how a three-mesh grid study can be used to estimate discretisation uncertainty with the GCI.

3. Discuss the requirements for a defensible CFD validation against experimental data.

4. Compare local linear uncertainty propagation, Monte Carlo sampling and surrogate-based approaches.

5. Develop a complete VVUQ workflow for an unsteady turbulent flow problem.

Analytical / Application Questions

1. Three successively refined meshes produce pressure-drop values of 120, 112 and 110 Pa. Explain what additional information is required before claiming asymptotic convergence and calculating a reliable GCI.

2. A CFD model agrees with an experiment for mean drag but predicts a shedding frequency 20% too low. Assess whether the model is validated for steady and unsteady design decisions.

3. A heat-transfer prediction has a small mesh uncertainty but a large sensitivity to inlet turbulence intensity and surface roughness. Recommend how the engineering team should allocate further verification, experiments and uncertainty-analysis effort.

22. Multiple Choice Questions with Answers

Question 1

Which statement best distinguishes validation from verification?

A. Validation checks whether residuals are below a prescribed tolerance.

B. Validation checks whether the implemented equations represent the intended physical system.

C. Validation is performed only with manufactured solutions.

D. Validation concerns only mesh quality.

Correct Answer: B. Validation checks whether the implemented equations represent the intended physical system.

Question 2

What is the principal purpose of the method of manufactured solutions?

A. To calibrate turbulence-model constants using experimental data.

B. To generate realistic industrial boundary conditions.

C. To test code implementation against a deliberately prescribed exact solution.

D. To eliminate modelling uncertainty.

Correct Answer: C. To test code implementation against a deliberately prescribed exact solution.

Question 3

A small residual history alone is insufficient evidence of convergence because it may not demonstrate that

A. the fluid properties are constant.

B. the quantity of interest has stopped changing.

C. the mesh contains tetrahedral cells.

D. the experiment has been calibrated.

Correct Answer: B. the quantity of interest has stopped changing.

Question 4

The grid-convergence index is primarily intended to estimate

A. experimental bias.

B. turbulence-model inadequacy.

C. grid-related discretisation uncertainty.

D. uncertainty in fluid density.

Correct Answer: C. grid-related discretisation uncertainty.

Question 5

Richardson extrapolation is most defensible when

A. solutions are obtained on unrelated meshes with different boundary conditions.

B. the results lie in an asymptotic refinement range with a consistent convergence trend.

C. only one mesh is available.

D. the solution changes non-monotonically because of regime switching.

Correct Answer: B. the results lie in an asymptotic refinement range with a consistent convergence trend.

Question 6

Which practice strengthens experimental validation?

A. Tuning model constants using the same data used for validation.

B. Comparing results at unmatched flow conditions.

C. Reporting experimental and computational uncertainties together.

D. Reporting only the average error over all outputs.

Correct Answer: C. Reporting experimental and computational uncertainties together.

Question 7

A parameter can dominate output uncertainty even with moderate sensitivity when it has

A. a very narrow uncertainty distribution.

B. a large range of uncertainty.

C. no physical interpretation.

D. a zero covariance with all other parameters.

Correct Answer: B. a large range of uncertainty.

Question 8

For an unsteady vortex-shedding simulation, which additional comparison is especially important beyond mean drag?

A. Mesh file size.

B. Shedding frequency and phase or spectral behaviour.

C. Number of colour levels in a contour plot.

D. Compiler version only.

Correct Answer: B. Shedding frequency and phase or spectral behaviour.

Question 9

If successive mesh results oscillate rather than approach the solution monotonically, the analyst should

A. apply Richardson extrapolation without qualification.

B. assume the finest mesh is exact.

C. investigate whether the assumptions of the convergence estimate are violated.

D. replace all experimental data.

Correct Answer: C. investigate whether the assumptions of the convergence estimate are violated.

Question 10

Which statement about a surrogate model in uncertainty quantification is correct?

A. It requires no verification because it is inexpensive.

B. It can reduce repeated CFD cost but must be checked over its intended parameter range.

C. It removes model-form uncertainty automatically.

D. It is valid outside the training domain by definition.

Correct Answer: B. It can reduce repeated CFD cost but must be checked over its intended parameter range.