Parallel harmonic balance method for analysis of nonlinear mechanical systems

Mechanical vibration analysis and modelling are essential tools used in the design of various mechanical components and structures. In the case of turbine engine design specifically, the ability to accurately predict vibration of various parts is crucial to ensure their safe operation while maintaining efficiency. As the designs become increasingly complex and margins for errors get smaller, high fidelity numerical vibration models are necessary for their analysis. Research of parallel algorithms has progressed significantly in the last decades, thanks to the exponential growth of the world's available computational resources. This work explores the possibilities for parallel implementations for solving large scale nonlinear vibration problems. A C++ code using MPI was developed to validate these implementations in practice. The harmonic balance method is used in combination with finite elements discretisation and applied to an elastic body with the Green-Lagrange nonlinear model for large deformations. A parameter continuation scheme using a predictor-corrector approach is included to compute frequency response functions. A Newton-Raphson solver is used to solve the bordered nonlinear system of equations in the frequency domain. Three different parallel algorithms for solving the linearised problem in each Newton iteration are analysed - a sparse direct solver (using MUMPS library), GMRES (using PETSc library) and an inhouse implementation of FETI. The performance of the solvers is analysed using beam testcases and a fan blade geometry. Scalability of MUMPS and the FETI solver is assessed. Full nonlinear frequency response functions with turning points are also computed. Use of artificial coarse space and preconditioning in FETI is discussed as it greatly impacts convergence properties of the solver. The presented parallel linear solvers show promising scalability results and an ability to solve nonlinear systems of several million degrees of freedom.Open Acces

Parallel Algorithms for Time and Frequency Domain Circuit Simulation

As a most critical form of pre-silicon verification, transistor-level circuit simulation is an indispensable step before committing to an expensive manufacturing process. However, considering the nature of circuit simulation, it can be computationally expensive, especially for ever-larger transistor circuits with more complex device models. Therefore, it is becoming increasingly desirable to accelerate circuit simulation. On the other hand, the emergence of multi-core machines offers a promising solution to circuit simulation besides the known application of distributed-memory clustered computing platforms, which provides abundant hardware computing resources. This research addresses the limitations of traditional serial circuit simulations and proposes new techniques for both time-domain and frequency-domain parallel circuit simulations. For time-domain simulation, this dissertation presents a parallel transient simulation methodology. This new approach, called WavePipe, exploits coarse-grained application-level parallelism by simultaneously computing circuit solutions at multiple adjacent time points in a way resembling hardware pipelining. There are two embodiments in WavePipe: backward and forward pipelining schemes. While the former creates independent computing tasks that contribute to a larger future time step, the latter performs predictive computing along the forward direction. Unlike existing relaxation methods, WavePipe facilitates parallel circuit simulation without jeopardizing convergence and accuracy. As a coarse-grained parallel approach, it requires low parallel programming effort, furthermore it creates new avenues to have a full utilization of increasingly parallel hardware by going beyond conventional finer grained parallel device model evaluation and matrix solutions. This dissertation also exploits the recently developed explicit telescopic projective integration method for efficient parallel transient circuit simulation by addressing the stability limitation of explicit numerical integration. The new method allows the effective time step controlled by accuracy requirement instead of stability limitation. Therefore, it not only leads to noticeable efficiency improvement, but also lends itself to straightforward parallelization due to its explicit nature. For frequency-domain simulation, this dissertation presents a parallel harmonic balance approach, applicable to the steady-state and envelope-following analyses of both driven and autonomous circuits. The new approach is centered on a naturally-parallelizable preconditioning technique that speeds up the core computation in harmonic balance based analysis. The proposed method facilitates parallel computing via the use of domain knowledge and simplifies parallel programming compared with fine-grained strategies. As a result, favorable runtime speedups are achieved

Autonomous Volterra algorithm for steady-state analysis of nonlinear circuits

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Model reduction and simulation of nonlinear circuits via tensor decomposition

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Reduced-order modeling of geometrically nonlinear rotating structures using the direct parametrisation of invariant manifolds

The direct parametrisation method for invariant manifolds is a nonlinear reduction technique which derives nonlinear mappings and reduced-order dynamics that describe the evolution of dynamical systems along a low-dimensional invariant-based span of the phase space. It can be directly applied to finite element problems. When the development is performed using an arbitrary order asymptotic expansion, it provides an efficient reduced-order modeling strategy for geometrically nonlinear structures. It is here applied to the case of rotating structures featuring centrifugal effect. A rotating cantilever beam with large amplitude vibrations is first selected in order to highlight the main features of the method. Numerical results show that the method provides accurate reduced-order models (ROMs) for any rotation speed and vibration amplitude of interest with a single master mode, thus offering remarkable reduction in the computational burden. The hardening/softening transition of the fundamental flexural mode with increasing rotation speed is then investigated in detail and a ROM parametrised with respect to rotation speed and forcing frequencies is detailed. The method is then applied to a twisted plate model representative of a fan blade, showing how the technique can handle more complex structures. Hardening/softening transition is also investigated as well as interpolation of ROMs, highlighting the efficacy of the method

SCALABLE INTEGRATED CIRCUIT SIMULATION ALGORITHMS FOR ENERGY-EFFICIENT TERAFLOP HETEROGENEOUS PARALLEL COMPUTING PLATFORMS

Integrated circuit technology has gone through several decades of aggressive scaling.It is increasingly challenging to analyze growing design complexity. Post-layout SPICE simulation can be computationally prohibitive due to the huge amount of parasitic elements, which can easily boost the computation and memory cost. As the decrease in device size, the circuits become more vulnerable to process variations. Designers need to statistically simulate the probability that a circuit does not meet the performance metric, which requires millions times of simulations to capture rare failure events. Recent, multiprocessors with heterogeneous architecture have emerged as mainstream computing platforms. The heterogeneous computing platform can achieve highthroughput energy efficient computing. However, the application of such platform is not trivial and needs to reinvent existing algorithms to fully utilize the computing resources. This dissertation presents several new algorithms to address those aforementioned two significant and challenging issues on the heterogeneous platform. Harmonic Balance (HB) analysis is essential for efficient verification of large postlayout RF and microwave integrated circuits (ICs). However, existing methods either suffer from excessively long simulation time and prohibitively large memory consumption or exhibit poor stability. This dissertation introduces a novel transient-simulation guided graph sparsification technique, as well as an efficient runtime performance modeling approach tailored for heterogeneous manycore CPU-GPU computing system to build nearly-optimal subgraph preconditioners that can lead to minimum HB simulation runtime. Additionally, we propose a novel heterogeneous parallel sparse block matrix algorithm by taking advantages of the structure of HB Jacobian matrices as well as GPU’s streaming multiprocessors to achieve optimal workload balancing during the preconditioning phase of HB analysis. We also show how the proposed preconditioned iterative algorithm can efficiently adapt to heterogeneous computing systems with different CPU and GPU computing capabilities. Extensive experimental results show that our HB solver can achieve up to 20X speedups and 5X memory reduction when compared with the state-of-the-art direct solver highly optimized for twelve-core CPUs. In nowadays variation-aware IC designs, cell characterizations and SRAM memory yield analysis require many thousands or even millions of repeated SPICE simulations for relatively small nonlinear circuits. In this dissertation, for the first time, we present a massively parallel SPICE simulator on GPU, TinySPICE, for efficiently analyzing small nonlinear circuits. TinySPICE integrates a highly-optimized shared-memory based matrix solver and fast parametric three-dimensional (3D) LUTs based device evaluation method. A novel circuit clustering method is also proposed to improve the stability and efficiency of the matrix solver. Compared with CPU-based SPICE simulator, TinySPICE achieves up to 264X speedups for parametric SRAM yield analysis without loss of accuracy

Optimal aeroelastic trim for rotorcraft with constrained, non-unique trim solutions

New rotorcraft configurations are emerging, such as the optimal speed helicopter and slowed-rotor compound helicopter which, due to variable rotor speed and redundant lifting components, have non-unique trim solution spaces. The combination of controls and rotor speed that produce the best steady-flight condition is sought among all the possible solutions. This work develops the concept of optimal rotorcraft trim and explores its application to advanced rotorcraft configurations with non-unique, constrained trim solutions. The optimal trim work is based on the nonlinear programming method of the generalized reduced gradient (GRG) and is integrated into a multi-body, comprehensive aeroelastic rotorcraft code. In addition to the concept of optimal trim, two further developments are presented that allow the extension of optimal trim to rotorcraft with rotors that operate over a wide range of rotor speeds. The first is the concept of variable rotor speed trim with special application to rotors operating in steady autorotation. The technique developed herein treats rotor speed as a trim variable and uses a Newton-Raphson iterative method to drive the rotor speed to zero average torque simultaneously with other dependent trim variables. The second additional contribution of this thesis is a novel way to rapidly approximate elastic rotor blade stresses and strains in the aeroelastic trim analysis for structural constraints. For rotors that operate over large angular velocity ranges, rotor resonance and increased flapping conditions are encountered that can drive the maximum cross-sectional stress and strain to levels beyond endurance limits; such conditions must be avoided. The method developed herein captures the maximum cross-sectional stress/strain based on the trained response of an artificial neural network (ANN) surrogate as a function of 1-D beam forces and moments. The stresses/strains are computed simultaneously with the optimal trim and are used as constraints in the optimal trim solution. Finally, an optimal trim analysis is applied to a high-speed compound gyroplane configuration, which has two distinct rotor speed control methods, with the purpose of maximizing the vehicle cruise efficiency while maintaining rotor blade strain below endurance limit values.Ph.D.Committee Chair: Dimitri N. Mavris; Committee Co-Chair: Daniel P Schrage; Committee Member: David A. Peters; Committee Member: Dewey H. Hodges; Committee Member: J.V.R. Prasa

Model Reduction and Simulation of Nonlinear Circuits via Tensor Decomposition

Abstract-Model order reduction of nonlinear circuits (especially highly nonlinear circuits), has always been a theoretically and numerically challenging task. In this paper we utilize tensors (namely, a higher order generalization of matrices) to develop a tensor-based nonlinear model order reduction (TNMOR) algorithm for the efficient simulation of nonlinear circuits. Unlike existing nonlinear model order reduction methods, in TNMOR high-order nonlinearities are captured using tensors, followed by decomposition and reduction to a compact tensor-based reducedorder model. Therefore, TNMOR completely avoids the dense reduced-order system matrices, which in turn allows faster simulation and a smaller memory requirement if relatively lowrank approximations of these tensors exist. Numerical experiments on transient and periodic steady-state analyses confirm the superior accuracy and efficiency of TNMOR, particularly in highly nonlinear scenarios

Comparison of reduction methods for finite element geometrically nonlinear beam structures

The aim of this contribution is to present numerical comparisons of model-order reduction methods for geometrically nonlinear structures in the general framework of finite element (FE) procedures. Three different methods are compared: the implicit condensation and expansion (ICE), the quadratic manifold computed from modal derivatives (MD), and the direct normal form (DNF) procedure, the latter expressing the reduced dynamics in an invariant-based span of the phase space. The methods are first presented in order to underline their common points and differences, highlighting in particular that ICE and MD use reduction subspaces that are not invariant. A simple analytical example is then used in order to analyze how the different treatments of quadratic nonlinearities by the three methods can affect the predictions. Finally, three beam examples are used to emphasize the ability of the methods to handle curvature (on a curved beam), 1:1 internal resonance (on a clamped-clamped beam with two polarizations), and inertia nonlinearity (on a cantilever beam)