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Simulation-based digital twins have emerged as a powerful tool for evaluating the mechanical response of bridges. As virtual representations of physical systems, digital twins can provide a wealth of information that complements traditional inspection and monitoring data. By incorporating virtual sensors and predictive maintenance strategies, they have the potential to improve our understanding of the behavior and performance of bridges over time. However, as bridges age and undergo regular loading and extreme events, their structural characteristics change, often differing from the predictions of their initial design. Digital twins must be continuously adapted to reflect these changes. In this article, we present a Bayesian framework for updating simulation-based digital twins in the context of bridges. Our approach integrates information from measurements to account for inaccuracies in the simulation model and quantify uncertainties. Through its implementation and assessment, this work demonstrates the potential for digital twins to provide a reliable and up-to-date representation of bridge behavior, helping to inform decision-making for maintenance and management.
A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparable coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples.
We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach.
We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach.
We present an efficient algorithm that finds a globally optimal solution to the 2D Free Flight Trajectory Optimization Problem (aka Zermelo Navigation Problem) up to arbitrary precision in finite time. The algorithm combines a discrete and a continuous optimization phase. In the discrete phase, a set of candidate paths that densely covers the trajectory space is created on a directed auxiliary graph. Then Yen’s algorithm provides a promising set of discrete candidate paths which subsequently undergo a locally convergent refinement stage. Provided that the auxiliary graph is sufficiently dense, the method finds a path that lies within the convex domain around the global minimizer. From this starting point, the second stage will converge rapidly to the optimum. The density of the auxiliary graph depends solely on the wind field, and not on the accuracy of the
solution, such that the method inherits the superior asymptotic convergence properties of the optimal control stage.
In the clinical cancer therapy of regional hyperthermia nonlinear perfusion effects inside and outside the tumor seem to play a not negligible role. A stationary model of such effects leads to a nonlinear Helmholtz term within an elliptic boundary value problem. The present paper reports about the application of a recently designed adaptive multilevel FEM to this problem. For several 3D virtual patients, nonlinear versus linear model is studied. Moreover, the numerical efficiency of the new algorithm is compared with a former application of an adaptive FEM to the corresponding instationary model PDE.
We consider a linear iterative solver for large scale linearly constrained quadratic minimization problems that arise, for example, in optimization with PDEs. By a primal-dual projection (PDP) iteration, which can be interpreted and analysed as a gradient method on a quotient space, the given problem can be solved by computing sulutions for a sequence of constrained surrogate problems, projections onto the feasible subspaces, and Lagrange multiplier updates. As a major application we consider a class of optimization problems with PDEs, where PDP can be applied together with a projected cg method using a block triangular constraint preconditioner. Numerical experiments show reliable and competitive performance for an optimal control problem in elasticity.
Adaptation of antenna profiles for control of MR guided hyperthermia (HT) in a hybrid MR-HT system
(2007)
In an aging society where the number of joint replacements rises, it is important to also increase the longevity of implants.
In particular hip implants have a lifetime of at most 15 years. This derives primarily from
pain due to implant migration, wear, inflammation, and dislocation, which is affected by
the positioning of the implant during the surgery. Current joint replacement practice uses
2D software tools and relies on the experience of surgeons. Especially the 2D tools fail to
take the patients’ natural range of motion as well as stress distribution in the 3D joint
induced by different daily motions into account.
Optimizing the hip joint implant position for all possible parametrized motions under the
constraint of a contact problem is prohibitively expensive as there are too many motions
and every position change demands a recalculation of the contact problem. For the
reduction of the computational effort, we use adaptive refinement on the parameter
domain coupled with the interpolation method of Kriging. A coarse initial grid is to be
locally refined using goal-oriented error estimation, reducing locally high variances. This
approach will be combined with multi-grid optimization such that numerical errors are
reduced.
We introduce a novel adaptive Gaussian Process Regression (GPR) methodology for efficient construction of surrogate models for Bayesian inverse problems with expensive forward model evaluations. An adaptive design strategy focuses on optimizing both the positioning and simulation accuracy of training data in order to reduce the computational cost of simulating training data without compromising the fidelity of the posterior distributions of parameters. The method interleaves a goal-oriented active learning algorithm selecting evaluation points and tolerances based on the expected impact on the Kullback-Leibler divergence of surrogated and true posterior with a Markov Chain Monte Carlo sampling of the posterior. The performance benefit of the adaptive approach is demonstrated for two simple test problems.
Adaptive Gaussian Process Regression for Efficient Building of Surrogate Models in Inverse Problems
(2023)
In a task where many similar inverse problems must be solved, evaluating costly simulations is impractical. Therefore, replacing the model y with a surrogate model y(s) that can be evaluated quickly leads to a significant speedup. The approximation quality of the surrogate model depends strongly on the number, position, and accuracy of the sample points. With an additional finite computational budget, this leads to a problem of (computer) experimental design. In contrast to the selection of sample points, the trade-off between accuracy and effort has hardly been studied systematically. We therefore propose an adaptive algorithm to find an optimal design in terms of
position and accuracy. Pursuing a sequential design by incrementally appending the computational budget leads to a convex and constrained optimization problem. As a surrogate, we construct a Gaussian process regression model. We measure the global approximation error in terms of its impact on the accuracy of the identified parameter and aim for a uniform absolute tolerance, assuming that y(s) is computed by finite element calculations. A priori error estimates and a coarse estimate of computational effort relate the expected improvement of the surrogate model error to computational effort, resulting in the most efficient combination of sample point and evaluation tolerance. We also allow for improving the accuracy of already existing sample points by continuing previously truncated finite element solution procedures.
Generating simulated training data needed for constructing sufficiently accurate surrogate models to be used for efficient optimization or parameter identification can incur a huge computational effort in the offline phase. We consider a fully adaptive greedy approach to the computational design of experiments problem using gradient-enhanced Gaussian process regression as surrogates. Designs are incrementally defined by solving an optimization problem for accuracy given a certain computational budget. We address not only the choice of evaluation points but also of required simulation accuracy, both of values and gradients of the forward model.
Numerical results show a significant reduction of the computational effort compared to just position-adaptive and static designs as well as a clear benefit of including gradient information into the surrogate training.
The paper presents a particle method framework for resolving molecular dynamics. Error estimators for both the temporal and spatial discretization are advocated and facilitate a fully adaptive propagation. For time integration, the implicit trapezoidal rule is employed, where an explicit predictor enables large time steps. The framework is developed and exemplified in the context of the classical Liouville equation, where Gaussian phase-space packets are used as particles. Simplified variants are discussed shortly, which should prove to be easily implementable in common molecular dynamics codes. A concept is illustrated by numerical examples for one-dimensional dynamics in double well potential.
The highly localized dynamics of cardiac electrophysiology models call for adaptive simulation methods. Unfortunately, the overhead incurred by classical mesh adaptivity turns out to outweigh the performance improvements achieved by reducing the problem size. Here, we explore a different approach to adaptivity based on algebraic degree of freedom subset selection during spectral deferred correction sweeps, which realizes a kind of multirate higher order integration. Numerical experience indicates a significant performance increase compared to uniform simulations.