Probabilistic Design Optimization Research Articles

Probabilistic design optimization of rectangular skirted mudmats under combined loading in spatially variable soil

Natural soil exhibits inherent spatial variability due to complex physical and chemical formation processes, making it necessary to analyze, evaluation and design foundations using probabilistic method. In this study, the bearing capacity of rectangular skirted mudmats in spatially variable soil is investigated by three-dimensional random finite element analysis (3D-RFEA). Given the high computational cost of 3D-RFEA, an active learning probabilistic method based on Kriging surrogate model is proposed to predict the mean and coefficient of variation of bearing capacity factor of mudmats. Then, considering the effects of the skirt depth ratio of mudmats and the variation coefficient and scales of fluctuation of the undrained shear strength of soil, a prediction model of probabilistic failure envelope is proposed by combining the normalized deterministic failure envelope and the cumulative distribution function (CDF) of the uniaxial bearing capacity factor. Based on the prediction model of probabilistic failure envelope, an example of probabilistic design optimization of mudmats is finally given. Compared with the normative method, the design method based on the prediction model can design mudmats directly with the permissible failure probability.

A decoupled double-loop method with the adaptive allowable limits for probabilistic performance-based design optimization

In this study, inspired by the idea of the sequential optimization and reliability assessment (SORA) method, a new decoupled double-loop method for probabilistic performance-based design optimization (PPBDO) has been proposed. The highlight of the proposed method is the independence of this method from the type of problem and the limit state function; in such a way that it can be used in non-linear problems with multiple probabilistic constraints with the implicit form of the limit state function. However, it requires less computational effort than conventional double-loop methods. In each cycle of this method, deterministic performance-based design optimization was performed and then the reliability of the optimal design was evaluated. If the results of the reliability assessment are not desirable, the allowable limit of the constraints in the deterministic optimization problem will be changed based on the results of the reliability assessment. This process repeats until the results of the reliability assessment become desirable. The metaheuristic enhanced vibrating particle system algorithm and the Monte Carlo simulation method were used for optimization process and reliability analysis, respectively. To evaluate the efficiency of the proposed method, the optimization problem was defined so as to reduce the structural weight while meeting the acceptance criteria for performance-based design of steel moment frames according to ASCE 41-13. The PPBDO of three- and nine-story steel moment frames were investigated and the results showed that the proposed method was able to reduce the volume of calculations and direct the search path toward a reliable space.

Efficient Numerical Integration Algorithm of Probabilistic Risk Assessment for Aero-Engine Rotors Considering In-Service Inspection Uncertainties

Numerical integration methods have the characteristics of high efficiency and precision, making them attractive for aero-engine probabilistic risk assessment and design optimization of an inspection plan. One factor that makes the numerical integration method a suitable approach to in-service inspection uncertainties is the explicit derivation of the integration formula and integration domains. This explicit derivation ensures accurate characterization of a multivariable system’s failure risk evolution mechanism. This study develops an efficient numerical integration algorithm for probabilistic risk assessment considering in-service inspection uncertainties. The principle of probability conservation is applied to the transformation of the integration domain from the current flight cycle to the initial (N = 0) computational space. Consequently, the integration formula of failure probability is deduced, and a detailed mathematical demonstration of the proposed method is provided. An actual compressor disk is evaluated using the efficient numerical integration algorithm and the Monte Carlo simulation to validate the accuracy and efficiency of the proposed method. Results show that the time cost of the proposed algorithm is dozens of times lower than that of the Monte Carlo simulation, with a maximum relative error of 5%. Thus, the efficient numerical integration algorithm can be applied to failure analysis in the airworthiness design of commercial aero-engine components.

Cost- and Risk-Based Seismic Design Optimization of Nuclear Power Plant Safety Systems

Seismic analysis, design, and qualification of systems, structures, and components (SSCs) is a significant contributor to the capital cost of a nuclear power plant. To reduce capital costs of advanced nuclear power plants and make commercial nuclear energy more competitive, innovations are needed in their structural design and construction, and not just in the reactor core and associated systems. Seismic isolation has been identified as an important cost-cutting technology that enables standardization of equipment across various sites. This paper develops and demonstrates a cost- and risk-based seismic design optimization of a representative safety system in a nuclear power plant with the dual goals of minimizing overnight capital cost and meeting safety goals. The design optimization can also include component seismic isolation, in which case, the optimized design includes a set of equipment that needs to be seismically isolated to minimize capital cost. The open-source codes MASTODON and Dakota are used for seismic probabilistic risk assessment and design optimization, respectively. A generic nuclear facility with a safety system comprising SSCs that are common to nuclear power plants is considered for the demonstration of the design optimization and is assumed to be located at the Idaho National Laboratory site. Generic costs and seismic design cost functions are assumed for the SSCs of the safety system. The sum of the costs of the SSCs is minimized in the optimization process, while the risk of failure of the safety system is provided as a constraint. Results show that the optimization process reduces capital costs significantly while automatically prioritizing the safety of SSCs that contribute most to the risk of the safety system.

Extension of the single-nonlinear-mode theory by linear attachments and application to exciter-structure interaction

Under certain conditions, the dynamics of a nonlinear mechanical system can be represented by a single nonlinear modal oscillator. This holds, in particular, under external excitation near primary resonance or under self-excitation by negative damping of the respective mode. The properties of the modal oscillator can be determined by computational or experimental nonlinear modal analysis. The simplification to a single-nonlinear-mode model facilitates qualitative and global analysis, and substantially reduces the computational effort required for probabilistic methods and design optimization. Important limitations of this theory are that only purely mechanical systems can be analyzed and that the respective nonlinear mode has to be recomputed when the system’s structural properties are varied. With the theoretical extension proposed in this work, it becomes feasible to attach linear subsystems to the primary mechanical system, and to approximate the dynamics of this coupled system using only the nonlinear mode of the primary mechanical system. The attachments must be described by linear ordinary or differential-algebraic equations with time-invariant coefficient matrices. The attachments do not need to be of purely mechanical nature, but may contain, for instance, electric, magnetic, acoustic, thermal or aerodynamic models. This considerably extends the range of utility of nonlinear modes to applications as diverse as model updating or vibration energy harvesting. As long as the attachments do not significantly deteriorate the host system’s modal deflection shape, it is shown that their effect can be reduced to a complex-valued modal impedance and an imposed modal forcing term. In the present work, the proposed approach is computationally assessed for the analysis of exciter-structure interaction. More specifically, the force drop typically encountered in frequency response testing is revisited. A cantilevered beam with cubic spring and an attached electro-dynamical shaker serves as benchmark. The proposed approach shows excellent accuracy. Mainly the already known limitations of single-nonlinear-mode theory reappear. In particular, higher harmonics should not be too pronounced. In the transient case, the time scales of vibration and amplitude-phase modulation should be well separated, and the attachment dynamics should be in quasi-steady state.

A systematic and probabilistic approach for optimal design and on-site adaptive balancing of building central cooling systems concerning uncertainties

In current design practice, chillers and pumps are often oversized due to conservative consideration of uncertainties using safety factors to avoid the risk of undersizing, which often results in significant energy waste in operation. In recent years, probabilistic optimal design methods have been proposed for the components of cooling systems, enabling risk-based decision-making rather than sizing systems with safety margins to consider uncertainties. However, approaches for probabilistic optimal design and balancing of entire cooling systems are still absent. This article therefore presents a systematic approach of probabilistic optimal design and adaptive balancing for central cooling systems of buildings to minimize the impacts (energy waste and increased life-cycle cost) of oversizing in operation. The probabilistic optimal design considers both the uncertainties of design inputs and the flexibility of on-site adaptive balancing, while adaptive balancing enables flexible balancing to maximize energy saving according to characteristics of constructed systems. A case study is conducted to test and validate the proposed approach. Results show that significant cost reduction and energy saving were achieved for chillers and pumps, respectively, through the systematic approach of probabilistic design and adaptive balancing. Energy consumption of pumps was reduced by 41% through coordinating pump design with probabilistic chiller design.

Probabilistic optimal design of cleanroom air-conditioning systems facilitating optimal ventilation control under uncertainties

Buildings with spaces requiring strict temperature and humidity controls, such as pharmaceutical cleanrooms and semiconductor/microchip factories, have been growing very quickly in terms of total floor area and energy consumption. In such buildings, much of the energy is unnecessarily wasted due to the incoordination of system design and operation/control, especially under “off-design” and ever-changing ambient and load conditions. This paper, therefore, proposes a probabilistic optimal design method for cleanroom air-conditioning systems facilitating optimal ventilation control under uncertainties. To consider the effects of asynchronous loads in different zones/spaces with reduced computation demand, a probabilistic diversity factor method is proposed which is a simplified method to quantify the effects of uncertainties of space load diversities in multiple zones/spaces using diversity factors. The proposed design method is implemented and validated in the design optimization of air-conditioning systems for implementing four different ventilation control strategies considering possible and uncertain off-design conditions. The energy and economic performance as well as service satisfaction of the air-conditioning systems are also evaluated and compared. Results show that the proposed design method can obtain the optimal air-conditioning systems with minimum life-cycle cost and superior satisfaction of service.

Analysis and Control of Stochastic Systems Using Semidefinite Programming Over Moments

This technical note develops a unified methodology for probabilistic analysis and optimal control design for jump diffusion processes defined by polynomials. The statistical moments of these systems can be described by a system of linear ordinary differential equations. Typically, however, the low-order moments depend on higher order moments, thus requiring an infinite system of equations to compute any moment exactly. Here, we develop a methodology for bounding statistical moments by using the higher order moments as inputs to an auxiliary convex optimal control problem with semidefinite constraints. For steady-state problems, the auxiliary optimal control problem reduces to a static semidefinite program. The method applies to both controlled and uncontrolled stochastic processes. For stochastic optimal control problems, the method gives bounds on achievable performance and can be used to compute approximately optimal solutions. For uncontrolled problems, both upper and lower bounds on desired moments can be computed. While the accuracy of most moment approximations cannot be quantitatively characterized, our method guarantees that the moment of interest is between the computed bounds.

Probabilistic optimal design and on-site adaptive commissioning of building air-conditioning systems concerning uncertainties

Sizing of building air-conditioning systems is a critical issue in design practice concerning the building energy consumption in operation and risk of being undersized. In current practice, chillers and pumps are often oversized due to the rough consideration of uncertainties using safety factor to avoid the risk of being undersized, which results in significant energy waste in operation. In addition, the current design and commissioning do not provide means and flexibility for the air-conditioning systems to minimize their energy consumption when the systems are found oversized in operation. This paper presents a novel design and commissioning approach, consisting of probabilistic optimal design and on-site adaptive commissioning, for building air-conditioning systems to maximize their energy savings in operation. The probabilistic optimal design of an air-conditioning system involves two parts. One is probabilistic optimal design of chillers considering uncertainties, one is probabilistic optimal design of water circulation system considering uncertainties and the flexibility of on-site adaptive commissioning. Monte Carlo simulation is used to quantify uncertainties in system design and operation process. The on-site adaptive commissioning method has alternative commissioning schemes developed to maximize the energy saving based on the actual situation. A case study is performed to test and validate this new design and commissioning approach. Results show that about 20% energy saving could be achieved when the system is oversized by 20%, compared to conventional design and commissioning methods.

Probabilistic design optimization of wind turbine gear transmission system based on dynamic reliability

In this paper, a probabilistic design optimization method for wind power gear drive system has been put forward based on the dynamic reliability. A pure torsional dynamic model of 1.5 MW wind turbine gear transmission system is developed which is based on lumped parameter method with incorporation of gear meshing stiffness and transmission error. By taking the wind load and the gear parameters as random variables, the variation in dynamic reliability of the system with time is obtained. The probabilistic optimization design model of the wind turbine gear transmission system with minimum displacement and volume is built. The parameters affecting the dynamic performance and reliability are taken as optimization design variables. The system reliability, the effects of random parameters on the reliability, the normal meshing condition and gear strength are taken as constraint conditions. We choose the gear transmission system of a 1.5 MW wind turbine for the system optimization and experimental verification. The results show that the system optimization based on dynamic reliability can reduce the volume of the system by 17.94 % and the system reliability is also guaranteed. Moreover, the vibration of each component of the system is reduced which is beneficial for smooth operation of the gear transmission system of wind turbine.

Improving parameter estimation with an efficient sequential probabilistic collocation-based optimal design method

Accurate estimation of soil hydraulic parameters is essential for modeling of water flow and solute transport in unsaturated soil. Ensemble-based methods have been proposed to provide the optimal monitoring (data-collection) strategy to improve hydraulic parameter estimation. To guarantee the performance, however, these methods usually require relatively large ensemble sizes, leading to high computational cost. To this end, we develop an efficient sequential probabilistic collocation-based optimal design (SPCOD) method, which integrates the adaptive ANOVA (analysis of variance)-based probabilistic collocation-based Kalman filter with the sequential optimal design. At the forecast step, the probabilistic collocation method is used to construct a reduced-order surrogate for system model with the adaptive ANOVA-based polynomial chaos expansion (PCE). At the design step, an information metric in the PCE form is derived to evaluate candidate monitoring strategies. The one with the maximum expected information metric value is chosen as the optimal. At the analysis step, the PCE coefficients are sequentially updated by a square-root Kalman filter with the real-time measurements obtained from the optimal monitoring strategy. Two synthetic numerical cases involving water flow and solute transport in the vadose zone are employed to illustrate the performance of the proposed method. The results show that: (1) compared with conventional fixed monitoring strategies, the optimal monitoring strategy designed by SPCOD provides more accurate parameter estimation and state prediction; (2) compared with our previously developed sequential ensemble-based optimal design method (Man et al., 2016b), the new approach provides a more robust and accurate optimal monitoring design with nearly the same computational cost.

Confidence-Driven Design Optimization Using Gaussian Process Metamodeling With Insufficient Data

To reduce the computational cost, surrogate models have been widely used to replace expensive simulations in design under uncertainty. However, most existing methods may introduce significant errors when the training data is limited. This paper presents a confidence-driven design optimization (CDDO) framework to manage surrogate model uncertainty for probabilistic design optimization. In this study, a confidence-based Gaussian process (GP) modeling technique is developed to handle the surrogate model uncertainty in system performance predictions by taking both the prediction mean and variance into account. With a target confidence level, the confidence-based GP models are used to reduce the probability of underestimating the probability of failure in reliability assessment. In addition, a new sensitivity analysis method is proposed to approximate the sensitivity of the reliability at the target confidence level with respect to design variables, and thus facilitate the CDDO framework. Three case studies are introduced to demonstrate the effectiveness of the proposed approach.

Probabilistic design optimization and simplified geotechnical risk analysis for large open pit excavations

A probabilistic geotechnical design optimization framework for large open pit excavations is proposed in the present work. An appropriate objective function is formulated in terms of normalized cost of excavation, while the overall slope angle constitutes the decision variable. Probability of slope's failure is computed via Monte Carlo simulations. Optimization curves identifying the optimum slope's angle are developed for different risk and volume factors. For illustrative purposes, a typical example of a surface lignite mine is analyzed and results are thoroughly discussed. The suggested methodology can be used as a complementary tool in the decision analysis process of similar projects.

Parameter Sensitivity Analysis and Probabilistic Optimal Design for the Main-Shaft Device of a Mine Hoist

As a key component of a mine hoist system, the main-shaft device bears most of the bending moment and torque. Thus, it is important to quantify the performance of the main-shaft device and ensure the reliability of the device. The paper aims to investigate the impact of uncertain structural parameters on the reliability of the main-shaft device. The probabilistic optimal design is performed based on the parameter sensitivity results. First, the design of experiments is adopted to realize the stochastic response analysis of the main-shaft device. Second, the moment-based saddlepoint approximation method is used to evaluate the strength reliability of the device considering uncertain parameters. Next, the reliability-based sensitivity is derived to investigate the parametric significance of random input variables. Finally, the probabilistic optimal design involving the reliability sensitivity is conducted. The results of the sensitivity indicate that the torque and the contact length of left bearing have relatively greater impact on the structural reliability than other variables of the main-shaft device.

Probabilistic optimization of engineering system with prescribed target design in a reduced parameter space

A novel probabilistic robust design optimization framework is presented using a Bayesian inference framework. The objective of the study is to obtain probabilistic descriptors of the system parameters conditioned on the user-prescribed target probability distributions of the output quantities of interest or figures of merit of a system. A criterion-based identification of a reduced important parameter space is performed from the typically high number of parameters modeling the stochastically parametrized physical system. The criterion can be based on sensitivity indices, design constraints or expert opinion or a combination of these. The posterior probabilities on the reduced or important parameters conditioned on prescribed target distributions of the output quantities of interest are derived using the Bayesian inference framework. The probabilistic optimal design proposed here offers the distinct advantage of prescribing probability bounds of the system performance functions around the optimal design points such that robust operation is ensured. The proposed method has been demonstrated with two numerical examples including the optimal design of a structural dynamic system based on user-prescribed target distribution for the resonance frequency of the system.

Parametric Design Analysis of Magnetic Sensor Based on Model Order Reduction and Reliability-Based Design Optimization

This paper presents a novel parametric design analysis of the magnetic sensor module used to find mover positions in linear motors. The proposed framework utilizes a computationally inexpensive proper orthogonal decomposition-dynamic mode decomposition-based model order reduction (MOR), coupled with a multiparameter moment matching method. The model is then utilized by a moment-based probabilistic design optimization method that considers the manufacturing uncertainties to find the optimal sensor design. A magnetic sensor module is designed for employing the proposed framework, and the results show that: 1) the proposed MOR technique is accurate; 2) the reliability analysis performed during the probabilistic design optimization can help to reduce the excessive non-compliance costs for the manufacturers; and 3) the combined framework is able to significantly expedite the optimal design search.

System probability of failure and sensitivity analyses of composite plates under low velocity impact

Probabilistic response of graphite epoxy composite plates under low velocity impact is studied modeling matrix cracking and delamination damage. Probabilistic design methodology of composite material based on progressive damage model is developed and implemented in a user defined subroutine to consider the scatter in material properties (Young’s moduli, Poisson’s ratio, ultimate strength, and fracture energy) and in initial velocity. Damage initiation and propagation failure due to matrix cracking and delamination are investigated for limiting criteria for the low velocity impact. The Gaussian process response surface method, is adopted to determine the system probability of failure and sensitivity analyses of the composite plates. The scatter in random parameters shows significant influence on the contact force and displacement histories. There is a probability of underestimation of the peak contact force and displacement by 28% and 44% respectively, if the scatter in the properties is not considered. The target reliability of the plate is achieved for an optimal combination of impactor masses and velocities. The sensitivity based probabilistic design optimization procedure is investigated to achieve better strength and lighter weight of composite for body armors.

Probabilistic optimal design concerning uncertainties and on-site adaptive commissioning of air-conditioning water pump systems in buildings

Sizing of air-conditioning water pump systems in buildings is a critical issue in design practice concerning the pump energy consumption in operation and risk of being undersized. As a result, significant energy is often wasted in operation due to oversizing to avoid the risk of being undersized. In current practice, throttling of commissioning valves are commonly adopted to push water flowrate (and pressure head) back to the design point no matter how much oversizing exists in a system. That partly mitigates the oversizing problem. This paper presents a novel approach consisting of probabilistic optimal design concerning uncertainties and on-site adaptive commissioning to further maximize energy savings of constant water flow pump systems. Minimized throttling is achieved by on-site adaptive commissioning, which reduces unnecessary pressure head and significant energy consumption. Pumps selected by the probabilistic optimal design can operate under both conventional design conditions and the projected possible off-design (oversized) conditions. The projection is based on the probability distribution of actual pressure head, which is estimated using Monte Carlo simulation by quantifying uncertainties in pressure loss calculation and system construction. Three case studies are conducted to test and validate this new design and commissioning approach. Results show that about 20% energy saving could be achieved, when the system is oversized by 20%, compared to conventional design and commissioning methods. The proposed approach also offers better energy performance in general compared to the designs all using variable speed pumps.

The probabilistic analysis and optimal design of a bevel gear transmission system with failure interaction

The probabilistic analysis and optimal design of a bevel gear transmission system with failure interaction

Probabilistic Failure of Graphite Epoxy Composite Plates Due to Low Velocity Impact

Stochastic finite-element analysis of composite plates due to low velocity impact (LVI) is studied, considering the material properties (Young's modulii, Poisson's ratio, strengths, and fracture energy) and initial velocity as random parameters. Damage initiation and propagation failure due to matrix cracking are investigated for safety criteria for the LVI. Progressive damage mechanics is employed to predict the stochastic dynamic response of the plates. The Gaussian process response surface method (GPRSM) is presently adopted to determine the probability of failure (Pf). There is a possibility of underestimation of the peak contact force and displacement by 10.7% and 11.03%, respectively, if the scatter in the properties is not considered. The sensitivity-based probabilistic design optimization procedure is investigated to achieve better strength and lighter weight of composite for body armors.