Non-probabilistic Reliability-based Topology Optimization Research Articles

Non-probabilistic reliability-based topology optimization for two-material structure based on convex set and bounded field mixed model

Non-probabilistic reliability-based topology optimization for two-material structure based on convex set and bounded field mixed model

Non-probabilistic reliability-based topology optimization (NRBTO) scheme for continuum structures based on the strength constraint parameterized level set method and interval mathematics

In this paper, a study on non-probabilistic reliability-based topology optimization (NRBTO) scheme for continuum structures based on the parameterized level set method (PLSM) is conducted. The P-norm is used to replace the local stress constraint with a single comprehensive stress constraint that approximates the maximum stress to optimize the structure model with minimum volume. In the interval-set model, the stress constraint is transformed into the non-probabilistic reliability-based scheme and the reliability is evaluated by the optimization feature distance (OFD). In the NRBTO scheme, the unknown-but-bounded (UBB) uncertainties of the material and external load of the optimized structure are considered. By interpolating the level set function (LSF) with the multiquadric (MQ) splines, the partial differential equation (PDE) is transformed into an ordinary differential equation (ODE). In addition, the interval parameter vertex method is employed to obtain the stress constraint interval, and the shape derivative and adjoint vector methods are used to obtain the evolution velocity field of LSF. The optimality criterion (OC) method is used to solve the optimization problem. Finally, three examples are given, and the optimization model of stiffness constraint is introduced, which not only verifies the applicability of the topology optimization scheme based on non-probabilistic reliability but also compares the difference between the topology optimization with strength constraint and the topology optimization with stiffness constraint.

Non-probabilistic reliability-based topology optimization against loading uncertainty field with a bounded field model

Non-probabilistic reliability-based topology optimization against loading uncertainty field with a bounded field model

An Efficient Sequential Strategy for Non-probabilistic Reliability-based Topology Optimization (NRBTO) of Continuum Structures with Stress Constraints

In the study, the sequential strategy is proposed for non-probabilistic reliability-based topology optimization (NRBTO) of continuum structures with stress constraints. Firstly, the qp-relaxation criterion is involved to circumvent the stress singularity. Then the deterministic topology optimization formulation by Stress-Influence-Function (SIF) approach is described. Uncertainty quantification analysis is conducted to determine the feasible bounds of the elemental stresses under unknown-but-bounded uncertainties. For safety reasons, the non-probabilistic reliability concept is introduced in the optimization formulation. The sequential strategy is proposed to handle the NRBTO problem and thus the original nested process of topology optimization and reliability analysis was decoupled, in which the update of design variables and the shift of the strength feature in the stress influence function are conducted alternately in the optimization process. The Performance Measure Approach (PMA) is applied to calculate the shift value of the strength feature and a shift value function is presented to avoid the numerical oscillations caused by the large variation of strength feature. Three numerical examples are eventually given to illustrate the applicability and validity of the developed sequential strategy for NRBTO with stress constraints.

Non-probabilistic reliability-based topology optimization (NRBTO) of continuum structures with displacement constraints via single-loop strategy

Non-probabilistic reliability-based topology optimization (NRBTO) of continuum structures with displacement constraints via single-loop strategy

Reliability-based topology optimization for heterogeneous composite structures under interval and convex mixed uncertainties

With the rapid development of intelligent manufacturing and smart design, the optimal design conception for composite laminates is arising more attentions in both academic and engineering fields. Note that most of the current researches focus on the deterministic laminate design whereas the uncertainty-oriented topological design scheme for heterogeneous composites are rarely considered. In view of this, this study will present a new layout optimization process which combines the hybrid uncertainty quantification analysis and non-probabilistic reliability-based topology optimization (NRBTO). Owing to the sample limitation of multi-source uncertainties in material properties and external loads, the non-statistical convex-theoretical set theory is first explored to determine displacement response bounds of multi-layer structures under anisotropic constitutive relationship. As the safety judgment, a novel reliability index corresponding to the local stiffness features is defined via interval and convex mixed uncertainties, and its explicit expression is further deduced by the set-interference principle. To ensure the robustness and convergence of the large-scale iteration in NRBTO framework, the adjoint equations are also analyzed for the design sensitivity updating and the gradient algorithm is involved as well. Several typical optimization cases are presented to demonstrate the validity and effectiveness of the developed methodology.

Non-probabilistic Reliability-based Topology Optimization (NRBTO) Scheme for Continuum Structures Based on the parameterized Level-Set method and Interval Mathematics

In this paper, a study on non-probabilistic reliability-based topology optimization (NRBTO) scheme for continuum structures based on the parameterized Level-Set method (PLSM) is conducted, in which the unknown-but-bounded (UBB) uncertainties of material and external loads are taken into account simultaneously. By interpolating the level set function (LSF) with the compactly supported radial basis functions (CSRBFs), the partial differential equation (PDE) is transformed into an ordinary differential equation (ODE). Based on the interval-set model, the displacement constraint is transformed into the non-probabilistic reliability-based scheme and the reliability is evaluated by the optimization feature distance (OFD). Moreover, the interval parametric vertex approach, the concept of shape derivative and the adjoint vector method are employed to obtain the sensitivity between the optimization model and the pseudo time to obtain the evolution velocity field of LSF. By utilizing the optimization criterion (OC) method, the optimization problem can be solved iteratively. To verify the validity and applicability of the proposed NRBTO method, three examples are presented, and numerical results show that taking the UBB uncertainties effects into account during the topology optimization may have a significant influence on the final structural configurations.

Topology optimization of structures considering local material uncertainties in additive manufacturing

Additive manufacturing provides more freedom for the design of structures but also exhibits prominent local uncertainties of material properties, which bring potential challenges. The performance of structures should depend not only on uncertain variations of material properties but also on the spatial occurrence frequency of the extreme material properties. This paper proposes a non-probabilistic reliability-based topology optimization algorithm by considering local material uncertainties in additive manufacturing. The whole design domain is divided into several uncertainty regions (URs), whose size is proportional to the spatial occurrence frequency of extreme material properties. Within each UR, these uncertain-but-bounded variations of materials are correlated by the multi-dimensional ellipsoid model. Then, the multi-ellipsoid model for all URs is established by considering the overall material uncertainties of the structure. Thereafter, a non-probabilistic reliability-based topology optimization (NRBTO) is proposed for minimizing structural volume against displacement constraints by considering material uncertainties during additive manufacturing. Several 2D and 3D examples are presented to illustrate the effectiveness of the proposed method. Compared with solutions resulting from the deterministic topology optimization (DTO), NRBTO provides conservative designs with a larger volume fraction due to material uncertainties. When smaller URs are assigned to indicate the high occurrence frequency of extreme material properties, the NRBTO design becomes even conservative. The extreme case is equivalent to the deterministic topology optimization using the lower bound material.

A novel approach of reliability-based topology optimization for continuum structures under interval uncertainties

Purpose The purpose of this paper is to propose a novel non-probabilistic reliability-based topology optimization (NRBTO) method for continuum structural design under interval uncertainties of load and material parameters based on the technology of 3D printing or additive manufacturing. Design/methodology/approach First, the uncertainty quantification analysis is accomplished by interval Taylor extension to determine boundary rules of concerned displacement responses. Based on the interval interference theory, a novel reliability index, named as the optimization feature distance, is then introduced to construct non-probabilistic reliability constraints. To circumvent convergence difficulties in solving large-scale variable optimization problems, the gradient-based method of moving asymptotes is also used, in which the sensitivity expressions of the present reliability measurements with respect to design variables are deduced by combination of the adjoint vector scheme and interval mathematics. Findings The main findings of this paper should lie in that new non-probabilistic reliability index, i.e. the optimization feature distance which is defined and further incorporated in continuum topology optimization issues. Besides, a novel concurrent design strategy under consideration of macro-micro integration is presented by using the developed RBTO methodology. Originality/value Uncertainty propagation analysis based on the interval Taylor extension method is conducted. Novel reliability index of the optimization feature distance is defined. Expressions of the adjoint vectors between interval bounds of displacement responses and the relative density are deduced. New NRBTO method subjected to continuum structures is developed and further solved by MMA algorithms.

A nonprobabilistic reliability–based topology optimization method of compliant mechanisms with interval uncertainties

SummaryOn account of the inevitable multisource uncertainty factors in compliant mechanisms, which seriously affect the accuracy of output motion, a nonprobabilistic reliability–based topology optimization (NRBTO) framework for compliant mechanisms with interval uncertainties is introduced. Combined with the solid isotropic material with penalization (SIMP) model and the set‐theoretical interval method, the uncertainty quantification analysis is conducted to obtain mathematical approximations and boundary laws of considered mean compliance. By normalization treatment of the limit‐state function, a new quantified measure of the nonprobabilistic reliability is then defined. The compliance‐based NRBTO design method ensures the output motion realizing its target value accurately considering the uncertainty factors. The sensitivities of the nonprobabilistic reliability index with respect to design variables are calculated by the adjoint vector method. Two engineering examples are eventually presented to illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques.

Truss layout design under nonprobabilistic reliability‐based topology optimization framework with interval uncertainties

SummaryA nonprobabilistic reliability‐based topology optimization (NRBTO) method for truss structures with interval uncertainties (or unknown‐but‐bounded uncertainties) is proposed in this paper. The cross‐sectional areas of levers are defined as design variables, while the material properties and external loads are regard as interval parameters. A modified perturbation method is applied to calculate structural response bounds, which are the prerequisite to obtain structural reliability. A deviation distance between the current limit state plane and the objective limit state plane, of which the expression is explicit, is defined as the nonprobabilistic reliability index, which serves as a constraint function in the optimization model. Compared with the deterministic topology optimization problem, the proposed NRBTO formulation is still a single‐loop optimization problem, as the reliability index is explicit. The sensitivity results are obtained from an analytical approach as well as a direct difference method. Eventually, the NRBTO problem is solved by a sequential quadratic programming method. Two numerical examples are used to testify the validity and effectiveness of the proposed method. The results show significant effects of uncertainties to the topology configuration of truss structures.

Novel methodology of Non-probabilistic Reliability-based Topology Optimization (NRBTO) for multi-material layout design via interval and convex mixed uncertainties

This paper proposes an efficient topology optimization strategy for seeking the optimal layout of multi-material structures with mixed uncertainties of interval and convexity. In conjunction with the safety criterion for local displacement and the extended power-law interpolation scheme for the material property, the task of the optimization problem is formulated as to minimize the total volume of different materials while satisfying reliability requirement. By introducing the set-theoretical convex method, uncertainty quantification analysis under mixture of interval and convex uncertainties is firstly conducted for exploring boundary rules of considered responses. Combined with the area-ratio principle and the shortest distance judgment, a novel reliability index with favorable mathematical characteristics is then defined. Moreover, for guarantee the computational efficiency as well as the iterative convergence, the adjoint-variable model is also established and the investigated minimization issue can be solved by a gradient-based optimization algorithm. Eventually, several numerical applications are given to demonstrate the validity and reasonability of the present topology optimization methodology.

Nonprobabilistic reliability oriented topological optimization for multi-material heat-transfer structures with interval uncertainties

This study presents a nonprobabilistic reliability-based topology optimization (NRBTO) framework that combines a multi-material interpolation model and interval mathematics to achieve an optimal layout design for heat-transfer structures under unknown but bounded (UBB) uncertainties. In terms of the uncertainty quantification (UQ) issue, the interval dimension-wise method (IDWM) based on set collocation theory is first proposed to effectively determine the bounds of nodal temperature responses. For safety reasons, the interval reliability (IR) index corresponding to the thermal constraint is defined, and then a new design policy, i.e., the strategy of nonprobabilistic reliability oriented topological optimization is established. To circumvent problems of large-scale variable updating in a multi-material topology optimization procedure, theoretical deductions of the design sensitivity analysis are further given based on the adjoint-vector criterion and the chain principle. The validity and feasibility of the developed methodology are eventually demonstrated by several application examples.

Non-probabilistic reliability-based topology optimization of continuum structures considering local stiffness and strength failure

This study presents a novel non-probabilistic reliability-based topology optimization (NRBTO) framework to determine optimal material configurations for continuum structures under local stiffness and strength limits. Uncertainty quantification (UQ) analysis under unknown-but-bounded (UBB) inputs is conducted to determine the feasible bounds of structural responses by combining a material interpolation model with stress aggregation function and interval mathematics. For safety reasons, improved interval reliability indexes that correspond to displacement and stress constraints are applied in topological optimization issues. Meanwhile, an adjoint-vector based sensitivity analysis is further discussed from which the gradient features between reliability measures and design variables are mathematically deduced, and the computational difficulties in large-scale variable updating can be effectively overcome. Numerical examples are eventually given to demonstrate the validity of the developed NRBTO methodology.

A non-probabilistic reliability-based topology optimization (NRBTO) method of continuum structures with convex uncertainties

This paper develops a non-probabilistic reliability-based topology optimization (NRBTO) framework for continuum structures under multi-dimensional convex uncertainties. Combined with the solid isotropic material with penalization (SIMP) model and the set-theoretical convex method, the uncertainty quantification (UQ) analysis is firstly conducted to obtain mathematical approximations and boundary laws of considered displacement responses. By normalization treatment of the limit-state function, a new quantified measure of the non-probabilistic reliability is then defined and further deduced by the principle of the hyper-volume ratio. For circumventing optimization difficulties arising from large-scale design variables, the adjoint vector scheme for sensitivity analysis of the reliability index with respect to design variables are discussed as well. Numerical applications eventually illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques.

Multiscale reliability-based topology optimization methodology for truss-like microstructures with unknown-but-bounded uncertainties

This paper introduces a new optimization framework to multiscale structures with unknown-but-bounded (UBB) uncertainties, where the optimal design is applied at two scales: the macroscale, i.e., the structural level, where the non-probabilistic reliability-based topology optimization (NRBTO) is conducted, and the microscale, i.e., the material level, where the non-probabilistic reliability-based design optimization (NRBDO) for truss-like cell dimensions is achieved. By combination of the homogenization method and interval mathematics, rules of macro–micro stiffness equivalence as well as interval bounds of macro displacement responses under specific micro structural settings can be first confirmed. For safety reasons, novel reliability definitions are respectively given for the double-level design, in which the offset factor with good differential properties is applied for the NRBTO and the matching factor derived from the improved rule of interval sorting is utilized for the NRBDO. To circumvent problems of large-scale variable updating in two-level optimization procedures, the adjoint-based vector in macro layout design as well as the particle swarm optimization (PSO) algorithm in micro parametric design are further discussed. The usage, effectiveness, and the superiority of the developed methodology are eventually demonstrated by two 3D engineering applications.

Non-probabilistic reliability-based topology optimization with multidimensional parallelepiped convex model

In this paper, a new non-probabilistic reliability-based topology optimization (NRBTO) method is proposed to account for interval uncertainties considering parametric correlations. Firstly, a reliability index is defined based on a newly developed multidimensional parallelepiped (MP) convex model, and the reliability-based topology optimization problem is formulated to optimize the topology of the structure, to minimize material volume under displacement constraints. Secondly, an efficient decoupling scheme is applied to transform the double-loop NRBTO into a sequential optimization process, using the sequential optimization & reliability assessment (SORA) method associated with the performance measurement approach (PMA). Thirdly, the adjoint variable method is used to obtain the sensitivity information for both uncertain and design variables, and a gradient-based algorithm is employed to solve the optimization problem. Finally, typical numerical examples are used to demonstrate the effectiveness of the proposed topology optimization method.

A novel method of non-probabilistic reliability-based topology optimization corresponding to continuum structures with unknown but bounded uncertainties

A new non-probabilistic reliability-based topology optimization (NRBTO) method for detailed design of continuum structures is proposed in this paper, in which the unknown but bounded (UBB) uncertainties existing in material and external loads are considered simultaneously. Based on the interval set-theoretic model, traditional displacement constraint is transformed into the non-probabilistic reliability format, in which a novel measuring index, namely, the optimization feature distance is defined by a combination of the area-ratio ideas. Thus, the optimal layout issue is converted by the lightweight design problem with displacement reliability constraints. Moreover, the interval parametric vertex approach and the adjoint vector method are also employed to obtain the sensitivity information between the reliability measurement and design variables. By utilizing the method of moving asymptotes (MMA), the optimization problem that is investigated can be solved iteratively. Two engineering examples are eventually presented to demonstrate the validity and effectiveness of the methodology that has been developed, and numerical results show that considering the UBB uncertainty effects during the whole procedure of topology optimization may have a significant influence on the final structural configurations.

Dynamic non-probabilistic reliability-based topology optimization of truss with uncertain-but-bounded parameters

In this paper, non-probabilistic reliability indices for frequency and static displacement constraints are analyzed based on the ellipse convex model of elastic modulus and mass density. The dynamic non-probabilistic reliability-based topology optimization model of a truss is built, where the cross-sectional areas and nodal topology variables are taken as design variables. The objective is to minimize the structural total mass. Constraints are imposed on static stresses and non-probabilistic reliability indices of static displacement and natural frequency. A genetic algorithm is used as the optimization method to find optimal solutions in the outer loop and an analysis method is adopted to seek non-probabilistic reliability index according to implicit forms of the limit state function in the inner loop. Results of numerical examples show that the optimal mass of a non-probabilistic reliability-based topology optimization is larger than that of the deterministic topology optimization and the optimal mass increases with the increase of the non-probabilistic reliability requirement in order to ensure structural safety.

Non-probabilistic Reliability Based Topology Optimization Design of Two-material Structures

Using the convex model description and the quantified definition of the non-probabilistic reliability,topology optimization of two-material structures considering uncertainties in material properties,geometry and loads is studied.Based on the extended penalization scheme of relative density,a continuous minimax optimization problem,which satisfies the material volume constraint and the reliability requirement,is presented for finding the united distribution of two candidate materials.The minimax problem is solved in the sequential approximate programming manner by embedding a direct iterative formula and the method of moving asymptotes(MMA).The computation cost is thus greatly decreased since the original problem is transformed into a series of deterministic ones.Numerical examples show that system uncertainties may have considerable effects on the optimal united layout of two-material structures.The results prove that a powerful approach for the topology design of two-material structures is furnished by the proposed numerical technique.