John T. Hwang

Paul Saves, Remi Lafage, Nathalie Bartoli et al.

The Surrogate Modeling Toolbox (SMT) is an open-source Python package that offers a collection of surrogate modeling methods, sampling techniques, and a set of sample problems. This paper presents SMT 2.0, a major new release of SMT that introduces significant upgrades and new features to the toolbox. This release adds the capability to handle mixed-variable surrogate models and hierarchical variables. These types of variables are becoming increasingly important in several surrogate modeling applications. SMT 2.0 also improves SMT by extending sampling methods, adding new surrogate models, and computing variance and kernel derivatives for Kriging. This release also includes new functions to handle noisy and use multifidelity data. To the best of our knowledge, SMT 2.0 is the first open-source surrogate library to propose surrogate models for hierarchical and mixed inputs. This open-source software is distributed under the New BSD license.

Yeonsu Kim, Junhan Lee, Bingran Wang et al.

Forward uncertainty quantification in dynamical systems is challenging due to non-smooth or locally oscillating nonlinear behaviors. Spline dimensional decomposition (SDD) addresses such nonlinearity by partitioning input coordinates via knot placement, but its accuracy is highly sensitive to internal knot locations. Optimizing knots using sequential quadratic programming is effective, yet computationally expensive. We propose a computationally efficient, interpolation-based method for optimal knot selection in SDD. The method includes: (1) interpolating input-output profiles, (2) defining subinterval-based reference regions, and (3) selecting knots at maximum gradient points within each region. The resulting knot vector is then applied to SDD for accurate approximation of non-smooth and oscillatory responses. A modal analysis of a lower control arm shows that SDD with the proposed knots yields higher accuracy than SDD with uniformly or randomly spaced knots and a Gaussian process model. In this example, the proposed SDD achieves the lowest relative variance error (2.89%) for the first natural frequency distribution, compared to uniformly spaced knots (12.310%), randomly spaced knots (15.274%), and Gaussian process (5.319%). All surrogates are constructed using the same 401 simulation datasets, and errors are evaluated against a 2000-sample Monte Carlo simulation. Scalability and applicability are demonstrated through stochastic and reliability analyses of one- and three-dimensional benchmark functions, and a ten-dimensional lower control arm model. Results confirm that second-moment statistics and reliability estimates can be accurately obtained with only a few hundred function evaluations or finite element simulations.