Robust Design Optimization Formulation Research Articles

Probabilistic Digital Twin for Additive Manufacturing Process Design and Control

Abstract This paper proposes a detailed methodology for constructing an additive manufacturing (AM) digital twin for the laser powder bed fusion (LPBF) process. An important aspect of the proposed digital twin is the incorporation of model uncertainty and process variability. A virtual representation of the LPBF process is first constructed using a physics-based model. To enable faster computation required in uncertainty analysis and decision-making, the physics-based model is replaced by a cheaper surrogate model. A two-step surrogate model is proposed when the quantity of interest is not directly observable during manufacturing. The data collected from the monitoring sensors are used for diagnosis (of current part quality) and passed on to the virtual representation for model updating. The model updating consists of Bayesian calibration of the uncertain parameters and the discrepancy term representing the model prediction error. The resulting digital twin is thus tailored for the particular individual part being produced and is used for probabilistic process parameter optimization (initial, before starting the printing) and online, real-time adjustment of the LPBF process parameters, in order to control the porosity in the manufactured part. A robust design optimization formulation is used to minimize the mean and standard deviation of the difference between the target porosity and the predicted porosity. The proposed methodology includes validation of the digital twin in two stages. Validation of the initial model in the digital twin is performed using available data, whereas data collected during manufacturing are used to validate the overall digital twin.

A new tilt-arm transitioning unmanned aerial vehicle: Introduction and conceptual design

In this paper, a novel hybrid unmanned aerial vehicle (UAV) concept is developed. This UAV is capable of transitioning between VTOL, hover, and efficient (fixed-wing type) forward flight. The overall configuration comprises a blended-wing-body, with two rotor arms mounted at the two wing tips using span-wise shafts; the arms can rotate about the span-wise axis, and each contains two propellers at its two ends. A conceptual design automation framework is developed, comprising mass and balance analysis, aerodynamic analysis and optimization. Vortex Lattice Method (VLM) is used to perform the aerodynamic analysis. Furthermore, using wind distribution models, redundancy modeling, and probabilistic UAV airspeed constraints derived thereof, a robust design optimization formulation is presented to explore the mission envelop flexibility of this new hybrid UAV. Mixed-discrete Particle Swarm Optimization is used to identify optimum geometry and component choices. Multiple case studies are performed to separately maximize forward-flight range and hovering endurance, subject to various aerodynamic and geometric constraints. The range optimizations converge to distinct designs under calm, windy, and stormy scenarios while offering promising flight ranges going from 40 km to 150 km, and the hovering optimization outcomes provide a 25 min hovering endurance subject to a 40 km round-trip flight.

Decoupling uncertainty quantification from robust design optimization

Robust design optimization (RDO) has been eminent in determining the optimal design of real-time complex systems under stochastic environment. Unlike conventional optimization, RDO involves uncertainty quantification which may render the procedure to be computationally intensive, if not prohibitive. In order to deal with such issues, an efficient approximation-based generalized RDO framework has been proposed. Since RDO formulation comprises of statistical terms of the performance functions, the proposed framework deals with approximation of those statistical quantities, rather than the performance functions. Consequently, the proposed framework allows transformation of the RDO problem to an equivalent deterministic one. As a result, unlike traditional surrogate-assisted RDO, the proposed framework yields desirable results in significantly less number of functional evaluations. For performing such response statistical approximation, two adaptive sparse refined Kriging-based computational models have been proposed. However, the generality of the proposed methodology allows any surrogate models to be employed within this framework, provided it is capable of capturing the functional non-linearity. Implementation of the proposed framework in three test examples and two finite element-based practical problems clearly illustrates its potential for further complicated applications.

Maximum length scale in density based topology optimization

The focus of this work is on two new techniques for imposing maximum length scale in topology optimization. Restrictions on the maximum length scale provide designers with full control over the optimized structure and open possibilities to tailor the optimized design for broader range of manufacturing processes by fulfilling the associated technological constraints. One of the proposed methods is based on combination of several filters and builds on top of the classical density filtering which can be viewed as a low pass filter applied to the design parametrization. The main idea is to construct band pass filter which restricts the appearance of very thin and very thick elements in the design. In combination with the robust design optimization formulation the methodology results in manufacturable designs without the need of any post processing. The second technique provides more strict control on the maximum design features and is developed with the help of morphological operators. The formulation relies on a small number of additional constraints. Both approaches are demonstrated on optimization problems in linear elasticity.

Probabilistic measures for assessing appropriateness of robust design optimization solutions

Robust design optimization (RDO) is a popular framework for addressing uncertainties in the design of engineering systems by considering different statistical measures, typically the mean and standard deviation of the system response. RDO can lead to a wide range of different candidate designs, establishing a different compromise between these competing objectives. This work introduces a new robustness measure, termed probability of dominance, for assessing the appropriateness of each candidate design. This measure is defined as the likelihood that a particular design will outperform the rival designs within a candidate set. Furthermore, a multi-stage implementation is introduced to facilitate increased versatility/confidence in the decision-making process by considering the comparison among smaller subsets within the initial larger set of candidate designs. For enhancing the robustness in these comparisons the impact of prediction errors, introduced to address potential differences between the real (i.e. as built) system and the numerical model adopted for it, is also addressed. This extends to proper modeling of the influence of the prediction error, including selection of its probability model, as well as evaluation of its impact on the probability of dominance and on the RDO formulation itself. Two illustrative examples are presented, the first considering the design of a tuned mass damper (TMD) for vibration mitigation of harmonic excitations and the second a topology optimization problem for minimum compliance. Extensive comparisons are presented in these two examples and the discussions demonstrate the power of the proposed approach for assessing the designs' robustness.

Neurocomputing strategies for solving reliability‐robust design optimization problems

PurposeThis paper, by taking randomness and uncertainty of structural systems into account aims to implement a combined reliability‐based robust design optimization (RRDO) formulation. The random variables to be considered include the cross section dimensions, modulus of elasticity, yield stress, and applied loading. The RRDO problem is to be formulated as a multi‐objective optimization problem where the construction cost and the standard deviation of the structural response are the objectives to be minimized.Design/methodology/approachThe solution of the optimization problem is performed with the non‐dominant cascade evolutionary algorithm with the weighted Tchebycheff metric, while the probabilistic analysis required is carried out with the Monte Carlo simulation method. Despite the computational advances, the solution of a RRDO problem for real‐world structures is extremely computationally demanding and for this reason neurocomputing estimations are implemented.FindingsThe obtained estimates with the neural network predictions are shown to be very satisfactory in terms of accuracy for performing this type of computation. Furthermore, the present numerical results manage to achieve a reduction in computational time up to four orders of magnitude, for low probabilities of violation, compared to the conventional procedure making thus feasible the reliability‐robust design optimization of realistic structures under probabilistic constraints.Originality/valueThe novel parts of the present work include the implementation of neurocomputing strategies in RRDO problems for reducing the computational cost and the comparison of the results given by RRDO and robust design optimization formulations, where the significance of taking into account probabilistic constraints is emphasized.

Vulnerability-based robust design optimization of imperfect shell structures

A stochastic vulnerability-based robust design procedure of isotropic shell structures possessing uncertain initial geometric as well as material and thickness properties that are modeled as random fields is assessed against conventional and reliability-based robust design procedures. The main idea of the vulnerability-based design philosophy is to achieve robust optimum designs while allowing designers to determine explicitly accepted probabilities that various performance objectives will not be exceeded, by introducing additional probabilistic (vulnerability) constraints. For this purpose, a stochastic finite element methodology is incorporated into the framework of an efficient two-objective robust design optimization formulation. This combined approach is then implemented in order to obtain optimum designs of an “imperfect” shell structure involving random geometric deviations from its perfect geometry as well as a spatial variability of its modulus of elasticity and thickness. Two-objective functions, the material volume of the structure and the coefficient of variation of the buckling load of the shell, are used for the description of the optimization problem, subject to deterministic, reliability and vulnerability constraints.

Designer’s preferences regarding equality constraints in robust design optimization

Robust design optimization (RDO) problems can generally be formulated by appropriately incorporating uncertainty into the corresponding deterministic optimization problems. Equality constraints in the deterministic problem need to be carefully formulated into the RDO problem because of the strictness associated with their feasibility. In this context, equality constraints have been generally classified into two types: (1) those that must be satisfied regardless of uncertainty, examples include physics-based constraints, such as F = ma, and (2) those that cannot be satisfied because of uncertainty, which are typically designer-imposed, such as dimensional constraints. This paper addresses the notion of preferred degree of satisfaction of deterministic equality constraints under uncertainty. Whether or not a particular equality constraint can be exactly satisfied depends on the statistical nature of the design variables that exist in the constraint and on the designer’s preferences. In this context, this paper puts forth three contributions. First, we develop a comprehensive classification of equality constraints in a way that is mutually exclusive and collectively exhaustive. Second, we present a rank-based matrix approach to interactively classify equality constraints, which systematically incorporates the designer’s preferences into the classification process. Third, we present an approach to incorporate the designer’s intra-constraint and inter-constraint preferences for designer-imposed constraints into the RDO formulation. Intra-constraint preference expresses how closely a designer wishes to satisfy a particular constraint; for example, in terms of its mean and standard deviation. A designer may express inter-constraint preference if satisfaction of a particular designer-imposed constraint is more important than that of another. We present an optimization formulation that incorporates the above discussed constraint preferences, which provides the designer with the means to explore design space possibilities. The formulation entails interesting implications in terms of decision making. Two engineering examples are provided to illustrate the practical usefulness of the developments proposed in this paper.

Reliability based robust design optimization of steel structures

In this work the uncertainty of a structural system is taken into account in the framework of a structural Reliability based Robust Design Optimization (RRDO) formulation where probabilistic constraints are incorporated into the robust design optimization formulation. A robust design optimization problem is formulated as a multi-criteria optimization problem. The Pareto front representing the solution of the RRDO problem is composed by designs with a state of robustness, since their performance is the least sensitive to the variability of the uncertain variables. The cross section dimensions together with other structural parameters, such as the modulus of elasticity, the yield stress and the applied loading, are considered as random variables. For the solution of the RRDO problem, the non-dominant Cascade Evolutionary Algorithm is employed combined with a weighted Tchebycheff metric.

Robust Performance-Based Design Optimization of Steel Moment Resisting Frames

The concept of performance-based design is considered in the framework of Robust Design Optimization (RDO) for the design of steel structures. The RDO problem is treated as a two-objective optimization problem where the initial construction cost and the variance of the maximum interstorey drift for the 10% in 50 years hazard level are considered as the problem objectives to be minimized. The structural performance is evaluated by means of the reliability demand and resistance methodology of the FEMA-350 guidelines in order to take into account both uncertainty and randomness in a consistent manner. The structure is designed to respond for different levels of seismic hazard levels with a desired confidence. The limit-state damage cost is used as a measure for the assessment of selected designs that have been obtained through the proposed RDO formulation. The non deterministic finite element problem encountered, is solved using the Monte Carlo Simulation method. The NSGA-II algorithm has been combined with the Evolution Strategies optimization algorithm for solving the two-objective optimization problem at hand.

Robust seismic design optimization of steel structures

Stochastic performance measures can be taken into account, in structural optimization, using two distinct formulations: robust design optimization (RDO) and reliability-based design optimization (RBDO). According to a RDO formulation, it is desired to obtain solutions insensitive to the uncontrollable parameter variation. In the present study, the solution of a structural robust design problem formulated as a two-objective optimization problem is addressed, where cross-sectional dimensions, material properties and earthquake loading are considered as random variables. Additionally, a two-objective deterministic-based optimization (DBO) problem is also considered. In particular, the DBO and RDO formulations are employed for assessing the Greek national seismic design code for steel structural buildings with respect to the behavioral factor considered. The limit-state-dependent cost is used as a measure of assessment. The stochastic finite element problem is solved using the Monte Carlo Simulation method, while a modified NSGA-II algorithm is employed for solving the two-objective optimization problem.

Structural optimization considering the probabilistic system response

In engineering problems, the randomness and uncertainties are inherent and the scatter of structural parameters from their nominal ideal values is unavoidable. In Reliability Based Design Optimization (RBDO) and Robust Design Optimization (RDO) the uncertainties play a dominant role in the formulation of the structural optimization problem. In an RBDO problem additional non deterministic constraint functions are considered while an RDO formulation leads to designs with a state of robustness, so that their performance is the least sensitive to the variability of the uncertain variables. In the first part of this study a metamodel assisted RBDO methodology is examined for large scale structural systems. In the second part an RDO structural problem is considered. The task of robust design optimization of structures is formulated as a multi-criteria optimization problem, in which the design variables of the optimization problem, together with other design parameters such as the modulus of elasticity and the yield stress are considered as random variables with a mean value equal to their nominal value. .

Multidisciplinary Robust Design Optimization of an Internal Combustion Engine

In this paper, we introduce a multidisciplinary robust design optimization formulation to evaluate uncertainty encountered in the design process. The formulation is a combination of the bi-level Collaborative Optimization framework and the multiobjective approach of the compromise Decision Support Problem. To demonstrate the proposed framework, the design of a combustion chamber of an internal combustion engine containing two subsystem analyses is presented. The results indicate that the proposed Collaborative Optimization framework for multidisciplinary robust design optimization effectively attains solutions that are robust to variations in design variables and environmental conditions.