In this work, we bridge standard adaptive mesh refinement and coarsening on scalable octree background meshes and robust unfitted finite element formulations for the automatic and efficient solution of large-scale nonlinear solid mechanics problems posed on complex geometries, as an alternative to standard body-fitted formulations, unstructured mesh generation and graph partitioning strategies. We pay special attention to those aspects requiring a specialized treatment in the extension of the unfitted h-adaptive aggregated finite element method on parallel tree-based adaptive meshes, recently developed for linear scalar elliptic problems, to handle nonlinear problems in solid mechanics. In order to accurately and efficiently capture localized phenomena that frequently occur in nonlinear solid mechanics problems, we perform pseudo time-stepping in combination with h-adaptive dynamic mesh refinement and rebalancing driven by a-posteriori error estimators. The method is implemented considering both irreducible and mixed (u/p) formulations and thus it is able to robustly face problems involving incompressible materials. In the numerical experiments, both formulations are used to model the inelastic behavior of a wide range of compressible and incompressible materials. First, a selected set of benchmarks are reproduced as a verification step. Second, a set of experiments is presented with problems involving complex geometries. Among them, we model a cantilever beam problem with spherical hollows distributed in a Simple Cubic array. This test involves a discrete domain with up to 11.7M Degrees Of Freedom solved in less than two hours on 3072 cores of a parallel supercomputer.
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A randomized Kaczmarz method was recently proposed for phase retrieval, which has been shown numerically to exhibit empirical performance over other state-of-the-art phase retrieval algorithms both in terms of the sampling complexity and in terms of computation time. While the rate of convergence has been studied well in the real case where the signals and measurement vectors are all real-valued, there is no guarantee for the convergence in the complex case. In fact, the linear convergence of the randomized Kaczmarz method for phase retrieval in the complex setting is left as a conjecture by Tan and Vershynin. In this paper, we provide the first theoretical guarantees for it. We show that for random measurements $\mathbf{a}_j \in \mathbb{C}^n, j=1,\ldots,m $ which are drawn independently and uniformly from the complex unit sphere, or equivalent are independent complex Gaussian random vectors, when $m \ge Cn$ for some universal positive constant $C$, the randomized Kaczmarz scheme with a good initialization converges linearly to the target solution (up to a global phase) in expectation with high probability. This gives a positive answer to that conjecture.
Soft robots are made of compliant and deformable materials and can perform tasks challenging for conventional rigid robots. The inherent compliance of soft robots makes them more suitable and adaptable for interactions with humans and the environment. However, this preeminence comes at a cost: their continuum nature makes it challenging to develop robust model-based control strategies. Specifically, an adaptive control approach addressing this challenge has not yet been applied to physical soft robotic arms. This work presents a reformulation of dynamics for a soft continuum manipulator using the Euler-Lagrange method. The proposed model eliminates the simplifying assumption made in previous works and provides a more accurate description of the robot's inertia. Based on our model, we introduce a task-space adaptive control scheme. This controller is robust against model parameter uncertainties and unknown input disturbances. The controller is implemented on a physical soft continuum arm. A series of experiments were carried out to validate the effectiveness of the controller in task-space trajectory tracking under different payloads. The controller outperforms the state-of-the-art method both in terms of accuracy and robustness. Moreover, the proposed model-based control design is flexible and can be generalized to any continuum robotic arm with an arbitrary number of continuum segments.
In this paper, a peridynamics-based finite element method (Peri-FEM) is proposed for the quasi-static fracture analysis, which is of the consistent computational framework with the classical finite element method (FEM). First, the integral domain of the peridynamics is reconstructed, and a new type of element called peridynamic element (PE) is defined. Although PEs are generated by the continuous elements (CEs) of classical FEM, they do not affect each other. Then the spatial discretization is performed based on PEs and CEs, and the linear equations about the nodal displacement are established according to the principle of minimum potential energy. Besides, the cracks are characterized as the degradation of the mechanical properties of PEs. Finally, the validity of the proposed method is demonstrated through numerical examples.
This paper presents fast non-sampling based methods to assess the risk for trajectories of autonomous vehicles when probabilistic predictions of other agents' futures are generated by deep neural networks (DNNs). The presented methods address a wide range of representations for uncertain predictions including both Gaussian and non-Gaussian mixture models to predict both agent positions and control inputs conditioned on the scene contexts. We show that the problem of risk assessment when Gaussian mixture models (GMMs) of agent positions are learned can be solved rapidly to arbitrary levels of accuracy with existing numerical methods. To address the problem of risk assessment for non-Gaussian mixture models of agent position, we propose finding upper bounds on risk using nonlinear Chebyshev's Inequality and sums-of-squares (SOS) programming; they are both of interest as the former is much faster while the latter can be arbitrarily tight. These approaches only require higher order statistical moments of agent positions to determine upper bounds on risk. To perform risk assessment when models are learned for agent control inputs as opposed to positions, we propagate the moments of uncertain control inputs through the nonlinear motion dynamics to obtain the exact moments of uncertain position over the planning horizon. To this end, we construct deterministic linear dynamical systems that govern the exact time evolution of the moments of uncertain position in the presence of uncertain control inputs. The presented methods are demonstrated on realistic predictions from DNNs trained on the Argoverse and CARLA datasets and are shown to be effective for rapidly assessing the probability of low probability events.
There has been a recent interest in imitation learning methods that are guaranteed to produce a stabilizing control law with respect to a known system. Work in this area has generally considered linear systems and controllers, for which stabilizing imitation learning takes the form of a biconvex optimization problem. In this paper it is demonstrated that the same methods developed for linear systems and controllers can be readily extended to polynomial systems and controllers using sum of squares techniques. A projected gradient descent algorithm and an alternating direction method of multipliers algorithm are proposed as heuristics for solving the stabilizing imitation learning problem, and their performance is illustrated through numerical experiments.
We discuss nonlinear model predictive control (NMPC) for multi-body dynamics via physics-informed machine learning methods. Physics-informed neural networks (PINNs) are a promising tool to approximate (partial) differential equations. PINNs are not suited for control tasks in their original form since they are not designed to handle variable control actions or variable initial values. We thus present the idea of enhancing PINNs by adding control actions and initial conditions as additional network inputs. The high-dimensional input space is subsequently reduced via a sampling strategy and a zero-hold assumption. This strategy enables the controller design based on a PINN as an approximation of the underlying system dynamics. The additional benefit is that the sensitivities are easily computed via automatic differentiation, thus leading to efficient gradient-based algorithms. Finally, we present our results using our PINN-based MPC to solve a tracking problem for a complex mechanical system, a multi-link manipulator.
The mathematical P2D model is a system of strongly coupled nonlinear parabolic-elliptic equations that describes the electrodynamics of lithium-ion batteries. In this paper, we present the numerical analysis of a finite element-implicit Euler scheme for such a model. We obtain error estimates for both the spatially semidiscrete and the fully discrete systems of equations, and establish the existence and uniqueness of the fully discrete solution.
Ensemble reinforcement learning (RL) aims to mitigate instability in Q-learning and to learn a robust policy, which introduces multiple value and policy functions. In this paper, we consider finding a novel but simple ensemble Deep RL algorithm to solve the resource consumption issue. Specifically, we consider integrating multiple models into a single model. To this end, we propose the \underline{M}inimalist \underline{E}nsemble \underline{P}olicy \underline{G}radient framework (MEPG), which introduces minimalist ensemble consistent Bellman update. And we find one value network is sufficient in our framework. Moreover, we theoretically show that the policy evaluation phase in the MEPG is mathematically equivalent to a deep Gaussian Process. To verify the effectiveness of the MEPG framework, we conduct experiments on the gym simulator, which show that the MEPG framework matches or outperforms the state-of-the-art ensemble methods and model-free methods without additional computational resource costs.
Exploration-exploitation is a powerful and practical tool in multi-agent learning (MAL), however, its effects are far from understood. To make progress in this direction, we study a smooth analogue of Q-learning. We start by showing that our learning model has strong theoretical justification as an optimal model for studying exploration-exploitation. Specifically, we prove that smooth Q-learning has bounded regret in arbitrary games for a cost model that explicitly captures the balance between game and exploration costs and that it always converges to the set of quantal-response equilibria (QRE), the standard solution concept for games under bounded rationality, in weighted potential games with heterogeneous learning agents. In our main task, we then turn to measure the effect of exploration in collective system performance. We characterize the geometry of the QRE surface in low-dimensional MAL systems and link our findings with catastrophe (bifurcation) theory. In particular, as the exploration hyperparameter evolves over-time, the system undergoes phase transitions where the number and stability of equilibria can change radically given an infinitesimal change to the exploration parameter. Based on this, we provide a formal theoretical treatment of how tuning the exploration parameter can provably lead to equilibrium selection with both positive as well as negative (and potentially unbounded) effects to system performance.
We present R-LINS, a lightweight robocentric lidar-inertial state estimator, which estimates robot ego-motion using a 6-axis IMU and a 3D lidar in a tightly-coupled scheme. To achieve robustness and computational efficiency even in challenging environments, an iterated error-state Kalman filter (ESKF) is designed, which recursively corrects the state via repeatedly generating new corresponding feature pairs. Moreover, a novel robocentric formulation is adopted in which we reformulate the state estimator concerning a moving local frame, rather than a fixed global frame as in the standard world-centric lidar-inertial odometry(LIO), in order to prevent filter divergence and lower computational cost. To validate generalizability and long-time practicability, extensive experiments are performed in indoor and outdoor scenarios. The results indicate that R-LINS outperforms lidar-only and loosely-coupled algorithms, and achieve competitive performance as the state-of-the-art LIO with close to an order-of-magnitude improvement in terms of speed.
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