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Why Use Brinkpy

Every result Brinkpy produces can, in principle, be reproduced with general scientific libraries and your own code. A first-order reliability method (FORM) analysis of a simple problem takes a few dozen lines with SciPy. Scripts like that are easier than ever to write, including with AI assistants.

The harder question is whether the number is right. A script that runs and returns a reasonable-looking reliability index is not the same as a result you can trust. In reliability analysis, a wrong answer rarely looks wrong. Brinkpy is built around that problem.

Checks on Every Run

Brinkpy checks the problem and the solution, not only the arithmetic:

  • Problem validation. Before any analysis, Brinkpy checks the stochastic variable definitions and the output of the limit state function, and reports how many function calls the check needed.
  • Derivative checks. When the limit state function returns its gradient or Hessian, Brinkpy compares them with finite differences before using them. The same applies to the derivatives of design objective and constraint functions. A wrong derivative is one of the most common errors in hand-written reliability code, and it can lead the solver to the wrong point without any sign of failure. See Derivative Checking.
  • Bounds handled as constraints. Variable bounds and truncated distributions are passed to the solver as constraints. Brinkpy reports which bounds are active at the solution and how strongly they affect the result. See Bounds and Truncation.
  • Diagnostic notes. Each run reports notes about anything that deserves attention, such as non-finite limit state function values, the cost of building an importance sampling proposal, or Markov chain Monte Carlo (MCMC) convergence statistics.

Many Methods, One Problem File

The same problem file runs with every method, under the same conventions for signs, transformations, and results:

  • FORM, and the second-order reliability method (SORM) with ten approximations, including the second-order saddlepoint approximation (SOSPA) and the inverse fast Fourier transform (IFFT) method.
  • Inverse FORM and Inverse SORM for reliability-targeted design values and partial factors.
  • Reliability-based design optimization with an objective function and constraints.
  • Monte Carlo simulation, with importance sampling proposals built by kernel density estimation, Gaussian mixtures, MCMC, or a normal distribution at the most probable point.
  • Plots of the probability density functions, the limit state function, and the reliability-based failure assessment diagram (RFAD).
  • Correlated and non-normal variables through the Nataf transformation.

Comparing methods on the same problem, for example FORM against several SORM approximations and Monte Carlo, takes a change of command flags or profile options, not new code.

Verification

Brinkpy is verified against published results. Every packaged example, and a larger set of problems from the literature, was checked against the results in the paper or book it comes from, and the results agree. These problems are part of the test suite of the Brinkpy API, which runs every time the API changes, so a change that breaks a known answer is caught before it is released.

The few differences from published results have a known cause, and in each case Brinkpy's result is the more accurate one:

  • Correlated variables in Ang and Tang. In the textbook examples with correlated non-normal variables, such as AT610, the book applies the correlation coefficients directly in standard normal space. Brinkpy adjusts them with the Nataf transformation, which accounts for how the correlation changes under the transformation to standard normal space.
  • The failure region of AU2. In the series system example of Au and Beck (1999), Brinkpy's importance sampling finds a part of the failure region that the published solution does not include, so its probability of failure, about 9.1e−7, is larger than the published 6.97e−7 (see MCMC Importance Sampling). When the variables are bounded to the region used in the paper, Brinkpy's estimate matches the published one within the variation expected from random sampling.

Reproducible Results

Each run saves a copy of the problem file, the profile with every setting used, a request ID, and the results in Excel, JSON, and PDF files. Another person can rerun the same problem with the same settings, or check exactly how a result was produced. See CLI Result Files.

Who Benefits Most

  • Students and instructors can focus on the methods and the results instead of debugging an implementation. The walkthrough examples use problems from standard textbooks.
  • Researchers can compare methods on the same problem and cite a fixed, versioned tool. See Citing Brinkpy.
  • Engineers who need reliability results but not a reliability code of their own get a checked implementation without writing one.