  The focus of this work is on the development of an error-driven isogeometric
framework, capable of automatically performing an adaptive simulation in the
context of second- and fourth-order, elliptic partial differential equations
defined on two-dimensional trimmed domains. The method is steered by an a
posteriori error estimator, which is computed with the aid of an auxiliary
residual-like problem formulated onto a space spanned by splines with single
element support. The local refinement of the basis is achieved thanks to the
use of truncated hierarchical B-splines. We prove numerically the applicability
of the proposed estimator to various engineering-relevant problems, namely the
Poisson problem, linear elasticity and Kirchhoff-Love shells, formulated on
trimmed geometries. In particular, we study several benchmark problems which
exhibit both smooth and singular solutions, where we recover optimal asymptotic
rates of convergence for the error measured in the energy norm and we observe a
substantial increase in accuracy per-degree-of-freedom compared to uniform
refinement. Lastly, we show the applicability of our framework to the adaptive
shell analysis of an industrial-like trimmed geometry modeled in the commercial
software Rhinoceros, which represents the B-pillar of a car.
