Implicit Differentiation Method Research Articles

Reliability analysis of fatigue crack growth in shallow shell structures using the Dual Boundary Element Method

This work presents a novel methodology for fatigue life reliability analysis using a newly developed Dual Boundary Element Method-based Implicit Differentiation Method (DBEM-IDM) in shallow shell structures. The proposed DBEM formulations evaluate stress intensity factors and fatigue life sensitivities in shallow shells considering geometrical variables and curvature. The IDM formulations, obtained through direct differentiation of the shallow shell DBEM formulations, uses the First-Order Reliability Method (FORM) to assess the reliability index and probability of failure of shallow shell structures with multiple cracks. Two numerical examples were simulated to illustrate the effectiveness of the proposed methodology. The first example examines a shallow shell with a centre hole and cracks subjected to various loads. The critical crack length required to ensure structural reliability and inspectability is determined. Fatigue reliability assessment is performed for the same example, employing limit state functions expressed in terms of fatigue life. FORM results are compared with the results of Monte Carlo Simulations (MCS), and provided a maximum difference of 3.67%. Parametric analyses are conducted to identify significant variables that affect reliability, and it was found that precise determination of design parameters is necessary to reduce the probability of failure, particularly for the Paris law constants. The second example, featuring more complex geometry, involved assessing fatigue reliability in a cylindrical shallow shell structure with multiple cracks. The results of this example indicate a 3.49% maximum difference between FORM and MCS. The results exhibit a low error compared to the MCS, and the inherent efficiency of the IDM-based FORM underscores its suitability for addressing uncertainty in reliability analysis.

Approaches to improve the convergence and efficiency of the implicit stress integration algorithm for the unified hardening model

To realize the numerical calculation of the unified hardening (UH) model for soils, the closest point projection method (CPPM) is used to do the stress integration in the finite element analysis. A package of approaches is proposed to improve the convergence and efficiency of the algorithm. An implicit differentiation method, which can greatly save the computational cost and storage space, is employed to calculate the second derivatives of the 3D yield function generalized by the transformed stress (TS) method. In addition, the denominator of the UH law is equal to 0 at the characteristic state and critical state, while the TS equation contains a square root and may be invalid for tensile stress. These singular problems are solved by introducing subtle treatments without changing the calculation results of the UH model. The consistent tangent operator corresponding to the stress integration can be expressed in a concise form. Finally, a series of simple loading tests and finite element simulations of the bearing capacity of a rigid circular foundation are conducted to evaluate the accuracy, convergence and efficiency of the proposed algorithm.

Implicit differentiation-based reliability analysis for shallow shell structures with the Boundary Element Method

A novel methodology for evaluating the response sensitivities of shallow shell structures using the Boundary Element Method (BEM) is presented in this work. The implicit derivatives of the BEM formulations for shallow shell structures, with respect to the geometrical variables, such as curvature and thickness, have been derived for the first time and incorporated into an Implicit Differentiation Method (IDM). The IDM is employed in conjunction with the First Order Reliability Method (FORM) to evaluate the reliability of shallow shell structures. The accuracy of the IDM formulation is first validated against an analytical solution, with results showing a maximum difference of only 2.61%. The IDM was later validated against the Finite Difference Method (FDM), with results showing a maximum difference of only 0.11%. The IDM was also found to be significantly more efficient than the FDM, requiring 35% less CPU time when calculating sensitivities. This is further compounded by the fact that, unlike the FDM, the IDM does not require a step size. A numerical example featuring a circular shallow shell is used to demonstrate the application of the IDM-based FORM for assessing structural reliability. The uncertainty in curvature is set as a variable for the purpose of investigating its impact on reliability. The results of the reliability index obtained from the IDM-FORM are compared to the results obtained from FDM-FORM and were found to be very similar. An analysis of sensitivity is conducted to identify the most significant variables affecting reliability. It is found that uncertainties in curvature, thickness, and applied pressure distribution parameters have the largest impact on structural reliability. To demonstrate how the IDM could be used in practice, it was employed as gradient-based optimisation procedure featuring shallow-shell structures. The IDM was found to be a very efficient and accurate alternative to existing methods for calculating structural response sensitivities.

Shape Optimisation of Assembled Plate Structures with the Boundary Element Method

A novel methodology is presented for performing sensitivity analyses of assembled plate structures using the Boundary Element Method (BEM). The main novelty of this work is that the exact implicit derivatives of the BEM formulations for assembled plate structures have been derived for the first time and incorporated into a newly developed Implicit Differentiation Method (IDM), enabling sensitivity analyses to be conducted for more complex and realistic structures in a more accurate, robust, and efficient manner than previous approaches. Three numerical examples are investigated to validate the derived exact implicit derivatives and to demonstrate how they could be used for a potential application involving the shape optimisation of a complex X-core structure from the canard of a Eurofighter Typhoon fighter jet. Results show that the newly developed IDM is more accurate, robust, and efficient when compared to alternative methodologies using derivatives obtained from methods such as the Finite Difference Method (FDM) and the Finite Element Method (FEM).

Manufacturing cost and reliability‐based shape optimization of plate structures

AbstractA novel methodology is presented for the manufacturing cost and reliability‐based optimization of plate structures with the boundary element method (BEM), with the aim of improving the accuracy, robustness, and efficiency of the optimization of aircraft structures. The BEM plate formulations with respect to plate thickness are derived for the first time, and used as part of an implicit differentiation method (IDM), enabling the full shape optimization of plate structures with the BEM. These implicit derivatives are validated against derivatives obtained from the finite difference method (FDM) and from an analytical solution. Results indicate that the IDM is more robust than the FDM and in excellent agreement with the analytical solution, and more accurate than the FDM for most of the step‐sizes investigated. To demonstrate the full shape optimization of plates with the newly developed IDM, a numerical example involving reliability‐based design optimization and manufacturing cost optimization is presented for a plate structure. Results show that the newly developed IDM is more efficient than the FDM when performing this optimization.

A dual boundary element based implicit differentiation method for determining stress intensity factor sensitivities for plate bending problems

A novel methodology for determining Stress Intensity Factor (SIF) sensitivities for plate bending problems using the Dual Boundary Element Method (DBEM) is presented. The direct derivatives of the DBEM integral equations for plate bending have been derived for the first time and are used as part of a DBEM-based Implicit Differentiation Method (IDM or DBEM-IDM) for calculating the sensitivities of SIFs to changes in different geometric parameters such as crack length and crack rotation angle. The SIFs and their sensitivities are calculated using the J-integral and the derivative of the J-integral respectively. A numerical example featuring a thick plate subjected to membrane, bending, and pressure loads is presented. In the first half of the numerical example, the SIF sensitivities from the IDM are compared with those obtained from the more common, but relatively crude, Finite Difference Method (FDM or DBEM-FDM). Results show that the IDM is a significantly more efficient and robust alternative to the FDM. The accuracy of the FDM showed significant dependence on the step size used, necessitating a time-consuming optimisation procedure to determine the optimal step size. Once this optimal step size was found, both methods provided very similar results. As part of the second half of the numerical example, a demonstration of one possible application of the SIF sensitivities from the IDM is presented. This involved carrying out reliability analyses using the First-Order Reliability Method (FORM) with a large number of design variables.

Design optimization for a simulation of rotating anisotropic viscoelastic porous structures using time-domain OQBEM

The main purpose of this paper is to propose a novel efficient design optimization technique based on the operational quadrature boundary element method (OQBEM) for a simulation study on the material distribution and influence of rotation on the optimal design of rotating anisotropic viscoelastic porous permeable structures. An implicit differentiation method was developed to calculate the design sensitivities of displacements and pore pressure with respect to design variables and the effects of rotation on these sensitivities are also discussed. The optimization problem is solved using the Moving Asymptotes Method (MAM) which requires the calculation of the sensitivities for objective function and constraints by adjoint variable method. The optimum shape design of an elliptical sandwich structure is used as the numerical example to verify the validity and accuracy of the proposed technique.

A multi-fidelity boundary element method for structural reliability analysis with higher-order sensitivities

A novel multi-fidelity modelling methodology for structural reliability analysis using the Boundary Element Method (BEM) with an Implicit Differentiation Method (IDM) is presented. Reliability analyses are conducted with methods such as Monte Carlo Simulation (MCS) and the First-Order Reliability Method (FORM). The higher-order sensitivities of the elastostatic Boundary Element Method equations with respect to changes in several geometric variables have been derived for the first time for use with the IDM for the purpose of conducting reliability analyses with the Second-Order Reliability Method (SORM), a more accurate alternative to FORM for problems with non-linear limit state functions. Multi-fidelity formulations involving the IDM have also been derived for the first time, making use of the metamodelling technique Kriging. The use of multi-fidelity modelling enables the creation of a model that has similar accuracy to a high-fidelity model, but with a computational cost similar to that of a low-fidelity model. By combining the accuracy of the IDM with the efficiency of multi-fidelity modelling the proposed methodology has the capability to be very effective when used for structural reliability analysis. The IDM is validated through a numerical example for which the analytical solution is known. A further two numerical examples featuring an I-beam section and a triangular support bracket with a large number of variables are also investigated. Results show that the employed multi-fidelity models were up to 6000 times faster in terms of CPU-time than the high-fidelity model, while also providing probabilities of failure that were up to 2225 times more accurate than the low-fidelity model. Overall, it has been shown that the use of the proposed IDM/multi-fidelity modelling methodology significantly improved the efficiency and accuracy of the above reliability analysis techniques when applied to complex problems involving a large number of random variables under high levels of uncertainty.

Shape design sensitivity and optimization for two-temperature generalized magneto-thermoelastic problems using time-domain DRBEM

ABSTRACTThe main purpose of this article is to propose an efficient shape optimization technique based on the time-domain dual reciprocity boundary element method (DRBEM) to solve the two-temperature generalized magneto-thermoelastic problems. An implicit differentiation method was used to compute the shape design sensitivity of an approximate DRBEM solution with respect to design variables. The feasible direction method was suggested and implemented for use with the golden-section search technique for minimizing the weight objective function while satisfying all constraints. The optimum shape design of a square plate with a hole is used as the numerical example to verify the accuracy of the proposed technique.

Use of the Saul’yev method for the digital simulation of chronoamperometry and linear sweep voltammetry at the ultramicrodisk electrode

The two-dimensional Saul’yev method of simulating processes at an ultramicrodisk electrode is compared with the fully implicit backward differentiation method started with a few backward implicit steps, and an alternating direction implicit method. 2D Saul’yev is convenient to program and although it is significantly slower in execution than the other two methods, it still executes in reasonable time, and yields equally good results with a suitable choice of discrete intervals, and despite its inherent propagation problem, and a certain restriction in the relationship of the spatial and temporal intervals. Saul’yev was implemented for the diffusion limited potential step experiment, as well as linear sweep voltammetry for a reversible system.

A Boundary Element Method for Structural Reliability

In this paper a sensitivity formulation using the Boundary Element Method (BEM) is presentedfor analysis of structural reliability problems. The sensitivity formulation is based on implicit differentiation method where the first and second order derivatives of the random variables are obtained directly by differentiation of the discretised boundary integral equation. The structural reliability is assessed using the Monte Carlo Method and FORM with BEM sensitivity parameters. A benchmark example is presented to demonstrate the accuracy and efficiency of the BEM for both Monte Carlo and Sensitivity based FORM approaches.

Probabilistic fracture mechanics by the boundary element method

In this paper, a new boundary element formulation is presented for probabilistic analysis of crack problems in context of linear elastic fracture mechanics. The method involves an implicit differentiation method for calculating fracture response derivatives with respect to random parameters and applies first-order reliability method to predicting the reliability of cracked structures. Direct differentiation is used to obtain the derivatives of fracture parameters with respect to crack size which is required for probabilistic analysis. Numerical examples are presented to illustrate the proposed method. The accuracy of the proposed method is demonstrated by comparing with either analytical solution or other established method.

A boundary element sensitivity formulation for contact problems using the implicit differentiation method

In this paper a sensitivity formulation using the boundary element method (BEM), for problems involving contact is presented. The proposed formulation is based on the implicit differentiation method (IDM), where the boundary integral equations are differentiated analytically with respect to the design variables. In the proposed formulation the design variables are defined in terms of the normal gap between the contact bodies. The analysis demonstrates that the proposed method is accurate and robust, as it does not resolve the whole system. The proposed method can be used for evaluating the sensitivities in any shape-optimisation problem involving contact.

Comparison of shape design sensitivity analysis formulations via material derivative-adjoint variable and implicit differentiation techniques for 3-D and 2-D curved boundary element

This paper reviews and compares two different approaches to continuum structural shape design sensitivity analysis: the material derivative-adjoint variable method and the implicit differentiation method using 3-D and 2-D BEM. Employing identical BEA models, curved element technology, numerical integration and equation solving techniques, the similarities and differences of both methods are highlighted. Significant theoretical and computational aspects associated with both methods are presented and accuracy and computational efficiency characteristics are demonstrated. Advantages and disadvantages of both techniques are identified. The characterization provided can aid in the prudent selection between these sensitivity analysis techniques, based on the characteristics of the proposed application.

Toward a Practical Design Optimization Tool

Abstract: Design optimization methodologies for sizing and shape optimal design have undergone an evolutionary process over the last two decades. The motivation behind the changes is to develop a methodology with no practical limitations on the nature and size of the problem that can be solved. In this research, a personal computer‐based shape optimal design system is developed with particular attention to robustness, generality, efficiency, and pre‐ and postprocessing. To achieve these objectives, the hybrid natural shape optimization approach is developed and used with adequate control over the velocity field matrix generation. User control over model creation and mesh generation makes it possible to minimize mesh distortion and the need for remeshing. Design sensitivity expressions using the implicit differentiation method are derived. It is shown that the analytical derivatives can be computed efficiently for a variety of functions. Numerical examples are solved to illustrate the developed methodology using a software system developed under the MS‐DOS operating system.

Optimum Young's Modulus Distribution in Tire Design

Abstract A new approach to determine the optimum compound characteristics in each tire material is developed and applied to improve various tire performances following the corresponding objective functions and constraints. In the new approach, optimization technology is combined with a finite element method so that the optimum compound characteristics in each tire material are obtained by minimizing an objective function, while satisfying the constraints. Hence, the targets of rubber characteristics in each material of a tire can be determined by the proposed approach to improve tire performance. When we optimize the compound characteristics in each element of a tire model, the number of design variables becomes more than a few hundred. Therefore, the sensitivity of the stiffness matrix in the optimization is obtained by a semi-analytical (or implicit differentiation) method and computational time is greatly reduced compared to the finite difference method. The new approach is applied and verified to improve the rolling resistance of a tire. Furthermore, we can obtain guidelines for a new topological tire structure by the Young's modulus optimization, considering that the rubber material should be removed from the initial tire design, if the optimum Young's modulus is almost zero at the element.

Dynamic Design Sensitivity Analysis Technique of Largely Deformed Elastoplastic Plate and Shell Structures.

The development of a dynamic structural optimization technique for largely deformed elastoplastic bodies can realize more rigorous structural design, such as an energy absorbing design for impact loading. In this study, an exact and direct sensitivity analysis technique of large dynamic deformation for the elastoplastic material governed by bilinear constitutive law is developed, based on the implicit differentiation method and Newmark method. A practical step-by-step calculation algorithm for the sensitivities of the large displacements and the elastoplastic stresses with respect to the design parameters is formulated. The validity of the sensitivity analysis technique suggested here is confirmed by implementing the numerical results.

Design Sensitivity Analysis Technique of Elastoplastic Bodies and Its Application to Shape Optimization.

The development of the shape optimization design technique of elastoplastic bodies can realize severe structural design such as limited material design, and is useful for the preliminary study of elastoplastic impact design. In this study, an exact and direct sensitivity analysis technique for the elastoplastic material governed by the bilinear constitutive law is developed, based on the implicit differentiation method. The sensitivity analysis technique suggested here is applied to determine optimum shapes of minimum weight under the elastoplastic stress constraints.

Sensitivity Analysis of Large Deformed Elastoplastic Shell Structures and Its Application to Structural Optimization.

An exact and direct sensitivity analysis technique of large deformation and buckling load for elastoplastic material governed by a bilinear constitutive law has been developed based on the implicit differentiation method. The sensitivity equation of the displacement increments has been derived from the Newton-Raphson iteration scheme, and a practical pseudo loading terms due to the sensitivities of the stiffness matrices and the internal load vector have been formulated by considering discontinuity of the displacement sensitivity at the yielding point. The validity of the formulation suggested here was confirmed by comparing the calculated results of several examples with those of the theory and the finite difference method. The sensitivity analysis technique suggested here was also applied to determine optimum shapes of minimum weight under the buckling load constraints

Design Sensitivity Analysis of Three-Dimensional Body by Boundary Element Method and Its Application to Shape Optimization.

A design sensitivity calculation technique based on the implicit differentiation method is fomulated for isoparametric boundary elements for three-dimensional(3D) shape optimization problems. The practical sensitivity equations for boundary displacements and stresses are derived, and efficiency and accuracy of the technique are compared with the semi-analytic method by implementing the sensitivity analysis of typical and basic shape design problems numerically. The sensitivity calculation technique is then applied to the minimum weight design problems of 3D bodies under stress constraints, such as the shape optimization of the ellipsoidal cavity in a cube and the connecting rod, where the Taylor series approximation, based on the boundary element sensitivity analysis at current design point, is adopted for the efficient implementation of the optimization.