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### Research Papers

J. Verif. Valid. Uncert. 2017;2(3):031001-031001-14. doi:10.1115/1.4037888.

This work is concerned with the use of Guderley's converging shock wave solution of the inviscid compressible flow equations as a verification test problem for compressible flow simulation software. In practice, this effort is complicated by both the semi-analytical nature and infinite spatial/temporal extent of this solution. Methods can be devised with the intention of ameliorating this inconsistency with the finite nature of computational simulation; the exact strategy will depend on the code and problem archetypes under investigation. For example, scale-invariant shock wave propagation can be represented in Lagrangian compressible flow simulations as rigid boundary-driven flow, even if no such “piston” is present in the counterpart mathematical similarity solution. The purpose of this work is to investigate in detail the methodology of representing scale-invariant shock wave propagation as a piston-driven flow in the context of the Guderley problem, which features a semi-analytical solution of infinite spatial/temporal extent. The semi-analytical solution allows for the derivation of a similarly semi-analytical piston boundary condition (BC) for use in Lagrangian compressible flow solvers. The consequences of utilizing this BC (as opposed to directly initializing the Guderley solution in a computational spatial grid at a fixed time) are investigated in terms of common code verification analysis metrics (e.g., shock strength/position errors, global convergence rates). For the examples considered in this work, the piston-driven initialization approach is demonstrated to be a viable alternative to the more traditional, direct initialization approach.

Commentary by Dr. Valentin Fuster
J. Verif. Valid. Uncert. 2017;2(3):031002-031002-14. doi:10.1115/1.4037887.

Validation assesses the accuracy of a mathematical model by comparing simulation results to experimentally measured quantities of interest. Model validation experiments emphasize obtaining detailed information on all input data needed by the mathematical model, in addition to measuring the system response quantities (SRQs) so that the predictive accuracy of the model can be critically determined. This article proposes a framework for assessing model validation experiments for computational fluid dynamics (CFD) regarding information content, data completeness, and uncertainty quantification (UQ). This framework combines two previously published concepts: the strong-sense model validation experiments and the modeling maturity assessment procedure referred to as the predictive capability maturity method (PCMM). The model validation experiment assessment requirements are captured in a table of six attributes: experimental facility, analog instrumentation and signal processing, boundary and initial conditions, fluid and material properties, test conditions, and measurement of system responses, with four levels of information completeness for each attribute. The specifics of this table are constructed for a generic wind tunnel experiment. Each attribute’s completeness is measured from the perspective of the level of detail needed for input data using direct numerical simulation of the Navier–Stokes equations. While this is an extraordinary and unprecedented requirement for level of detail in a model validation experiment, it is appropriate for critical assessment of modern CFD simulations.

Commentary by Dr. Valentin Fuster
J. Verif. Valid. Uncert. 2017;2(3):031003-031003-8. doi:10.1115/1.4038175.

Joint kinetic measurement is a fundamental tool used to quantify compensatory movement patterns in participants with transtibial amputation (TTA). Joint kinetics are calculated through inverse dynamics (ID) and depend on segment kinematics, external forces, and both segment and prosthetic inertial parameters (PIPS); yet the individual influence of PIPs on ID is unknown. The objective of this investigation was to assess the importance of parameterizing PIPs when calculating ID using a probabilistic analysis. A series of Monte Carlo simulations were performed to assess the influence of uncertainty in PIPs on ID. Multivariate input distributions were generated from experimentally measured PIPs (foot/shank: mass, center of mass (COM), moment of inertia) of ten prostheses and output distributions were hip and knee joint kinetics. Confidence bounds (2.5–97.5%) and sensitivity of outputs to model input parameters were calculated throughout one gait cycle. Results demonstrated that PIPs had a larger influence on joint kinetics during the swing period than the stance period (e.g., maximum hip flexion/extension moment confidence bound size: stance = 5.6 N·m, swing: 11.4 N·m). Joint kinetics were most sensitive to shank mass during both the stance and swing periods. Accurate measurement of prosthesis shank mass is necessary to calculate joint kinetics with ID in participants with TTA with passive prostheses consisting of total contact carbon fiber sockets and dynamic elastic response feet during walking.

Commentary by Dr. Valentin Fuster
J. Verif. Valid. Uncert. 2017;2(3):031004-031004-15. doi:10.1115/1.4038255.

A statistical approach for computational fluid dynamics (CFD) state-of-the-art (SoA) assessment is presented for specified benchmark test cases and validation variables, based on the combination of solution and N-version verification and validation (V&V). Solution V&V estimates the systematic numerical and modeling errors/uncertainties. N-version verification estimates the random simulation uncertainty. N-version validation estimates the random absolute error uncertainty, which is combined with the experimental and systematic numerical uncertainties into the SoA uncertainties and then used to determine whether or not the individual codes/simulations and the mean code are N-version validated at the $USoAi$ and USoA intervals, respectively. The scatter is due to differences in models and numerical methods, grid types, domains, boundary conditions, and other setup parameters. Differences between codes/simulations and implementations are due to myriad possibilities for modeling and numerical methods and their implementation as CFD codes and simulation applications. Industrial CFD codes are complex software with many user options such that even in solving the same application, different results may be obtained by different users, not necessarily due to user error but rather the variability arising from the selection of various models, numerical methods, and setup options. Examples are shown for ship hydrodynamics applications using results from the Seventh CFD Ship Hydrodynamics and Second Ship Maneuvering Prediction Workshops. The role and relationship of individual code solution V&V versus N-version V&V and SoA assessment are discussed.

Commentary by Dr. Valentin Fuster
J. Verif. Valid. Uncert. 2017;2(3):031005-031005-7. doi:10.1115/1.4038494.

Numerical codes are important in providing solutions to partial differential equations in many areas, such as the heat transfer problem. However, verification of these codes is critical. A methodology is presented in this work as an intrinsic verification method (IVM) to the solution to the partial differential equation. Derivation of the dimensionless form of scaled sensitivity coefficients is presented, and the sum of scaled sensitivity coefficients is used in the dimensionless form to provide a method for verification. Intrinsic verification methodology is demonstrated using examples of heat transfer problems in Cartesian and cylindrical coordinate. The IVM presented here is applicable to analytical as well as numerical solutions to partial differential equations.

Commentary by Dr. Valentin Fuster
J. Verif. Valid. Uncert. 2017;2(3):031006-031006-9. doi:10.1115/1.4038486.

Load-bearing mechanical structures like trusses face uncertainty in loading along with uncertainty in stress and strength, which are due to uncertainty in their development, production, and usage. According to the working hypothesis of the German Collaborative Research Center SFB 805, uncertainty occurs in processes that are not or only partial deterministic and can only be controlled in processes. The authors classify, compare, and evaluate four different direct methods to describe and evaluate the uncertainty of normal stress distribution in simple truss structures with one column, two columns, and three columns. The four methods are the direct Monte Carlo (DMC) simulation, the direct quasi-Monte Carlo (DQMC) simulation, the direct interval, and the direct fuzzy analysis with α-cuts, which are common methods for data uncertainty analysis. The DMC simulation and the DQMC simulation are categorized as probabilistic methods to evaluate the stochastic uncertainty. On the contrary, the direct interval and the direct fuzzy analysis with α-cuts are categorized as possibilistic methods to evaluate the nonstochastic uncertainty. Three different truss structures with increasing model complexity, a single-column, a two-column, and a three-column systems are chosen as reference systems in this study. Each truss structure is excited with a vertical external point load. The input parameters of the truss structures are the internal system properties such as geometry and material parameters, and the external properties such as magnitude and direction of load. The probabilistic and the possibilistic methods are applied to each truss structure to describe and evaluate its uncertainty in the developing phase. The DMC simulation and DQMC simulation are carried out with full or “direct” sample sets of model parameters such as geometry parameters and state parameters such as forces, and a sensitivity analysis is conducted to identify the influence of every model and state input parameter on the normal stress, which is the output variable of the truss structures. In parallel, the direct interval and the direct fuzzy analysis with α-cuts are carried out without altering and, therefore, they are direct approaches as well. The four direct methods are then compared based on the simulation results. The criteria of the comparison are the uncertainty in the deviation of the normal stress in one column of each truss structure due to varied model and state input parameters, the computational costs, as well as the implementation complexity of the applied methods.

Commentary by Dr. Valentin Fuster