Complex-step Derivative Approximation Research Articles

Hyperelastic Material Modeling Using First- and Second-Order Differential Operations with Complex Step Derivative Approximation

This study proposes a method for calculating hyperelastic materials’ stress and tangent modulus using a generalized complex-step derivative approximation. This method is accurate and requires only the operational code of the strain energy density function. Moreover, because it uses only standard complex operations in Fortran, no prior preparation is required, making the method convenient and practical. The effectiveness of the method was confirmed by applying it to the implementation of isotropic and anisotropic hyperelastic models.

Numerical approximation for algorithmic tangent moduli for nonlinear viscoelastic model with CSDA method

This work revisits the notion of complex step derivative approximation (CSDA) and presents its use in constitutive model of a class of nonlinear viscoelastic materials. The effectiveness of a CSDA is evaluated by putting it through a series of straightforward examples. After that, the idea of the CSDA is put to use in order to carry out a numerical evaluation of the algorithmic tangent moduli of a viscoelastic constitutive model. The performance of the constitutive models is evaluated through the use of three different numerical tests, and the results are compared to those that were achieved by the application of an analytical method. In comparison to other numerical differentiation techniques, It has been found that the CSDA scheme is the most computationally efficient and robust method of numerical differentiation, regardless of the size of the finite difference interval.

Robust and efficient implementation of finite strain generalized continuum models for material failure: Analytical, numerical, and automatic differentiation with hyper-dual numbers

Generalized continuum models for representing nonlinear material behavior including material failure in the finite strain regime are commonly formulated based on scalar elastic and dissipation potential functions. The evolution of stresses and internal variables, i.e., the material state, is governed by partial derivatives of the potential functions with respect to deformation and stress measures. Furthermore, for application of such models in implicit nonlinear finite element analysis tangent operators, consistent with the numerical integration algorithm, are required. In this work, we study analytical, numerical and automatic differentiation schemes for the implementation of coupled, generalized continuum models considering finite inelastic deformations and fracture. This includes the application of finite difference approximations, complex-step derivative approximations and automatic differentiation based on hyper-dual numbers. For the use of automatic differentiation, a semi-analytical split approach is introduced for increasing the computational efficiency. Based on a comprehensive 2D and 3D finite element study, we demonstrate the superior properties of automatic differentiation compared to numerical differentiation methods with regards to accuracy and robustness. Furthermore, the additional computational cost for automatic differentiation using the proposed split approach becomes negligible for large problems.

Implicit multiscale finite element analysis of polymer physics-based multiscale constitutive model for elastomers

This paper proposes a multiscale visco-hyperelastic constitutive model that can predict the material behavior bridging molecular properties to response at the continuum level and its integration within implicit finite element analysis. The proposed model incorporates all parameters having actual physical meaning obtained from molecular simulations based on the tube theory. Despite the advantages that the constitutive equation was determined from molecular network characteristics, applications of the model have been restricted to the simple shape of Finite Element (FE) models with a few cases of specific deformations, owing to its complexity and absence of the implicit formulation. Therefore, the application of the model to the 3D implicit FE analysis with a tangent stiffness was proposed by applying numerical differentiation with the complex step derivative approximation (CSDA) method. Combined with the Neo-Hookean hyperelasticity model, the results from the example analysis applied to the tensile specimen model were examined to manifest the visco-hyperelastic characteristics of elastomers under cyclic loading, stability of the analysis, and further discussion on the visco-hyperelastic constitutive model were proposed.

An implicit stress update algorithm for the plastic nonlocal damage model of concrete

The numerical implementation of advanced plastic damage models has been challenging due to the diverse computational difficulties caused by material nonlinearity. In this study, firstly, a plastic damage model is developed by a nonsmoothed Mohr–Coulomb yield criterion. The nonorthogonal flow rule without plastic potential function is employed to describe the dilatancy behavior of concrete. The stiffness degradation is considered through two tensile and compressive damage variables. Subsequently, an implicit stress update algorithm incorporating several advanced numerical methods is proposed to tackle difficulties arising in the numerical calculation of the plastic damage model. In the model implementation, the Mohr–Coulomb yield function with corners is replaced by a single smooth expression to ensure the stability of the numerical calculation. A line search method is used to solve nonlinear systems of integral stress equations to obtain better convergence than Newton method. The complex step derivative approximation is used to evaluate the consistent tangent operator to provide quadratic convergence speed of the global equilibrium iterations, thus avoiding tedious analytical derivative operations. The integral nonlocal method is also used to regularize the finite element solution. At the material point scale, the ability of the proposed model to describe the complex mechanical behavior of concrete is verified by utilizing nine sets of monotonic and cyclic loading and unloading tests. Finally, the algorithm and model’s performance is well tested by the failure analysis of five numerical examples. The code of this work will be freely available at https://github.com/zhouxin615/PD_model.

A robust stress update algorithm for elastoplastic models without analytical derivation of the consistent tangent operator and loading/unloading estimation

AbstractA robust and concise implicit stress integration algorithm of elastoplastic models is presented. It does not require the loading/unloading estimation and analytical derivation operation for the stress update. First, the elastoplastic stress update problem is recast into an unconstrained minimization problem by utilizing the smooth function to bypass the loading/unloading estimation. Then, the object problem is solved by the line search method instead of the Newton method for better convergence. The consistent tangent operator is evaluated by the complex step derivative approximation without the subtraction cancellation error, which provides the quadratic convergence rate of global iteration. The rationality of the numerical consistent tangent operator is validated by the one obtained by the analytical derivation. A recently developed non‐orthogonal elastoplastic (NEP) clay model is implemented using the new algorithm. The algorithm is confirmed through comparing the numerical solution and the analytical one for a cavity expansion problem. The algorithm performance is assessed based on a series of geotechnical boundary value problems. It is found that the new algorithm is more robust than the one employed by ABAQUS. The source code of the model implementation can be downloaded from https://github.com/zhouxin615.

Compact representation and identification of important regions of metal microstructures using complex-step convolutional autoencoders

In this study, we propose a complex-step convolutional autoencoder to identify the regions that are important in a metal microstructure for compact representation and secure sharing. Firstly, the architecture of a convolutional autoencoder is designed for the compact representation of microstructural images. The designed autoencoder achieved a high image compression ratio of 32 without loss of important information. Secondly, an in-home developed model agnostic sensitivity analysis using complex step derivative approximation is implemented on convolutional autoencoders to identify regions of the microstructure that are important for reconstruction. Finally, saliency maps that highlight the importance of pixels for reconstruction are generated for three grades of dual-phase structural steels. The saliency maps indicated secondary phase regions and grain boundaries are important for microstructure image reconstruction. The proposed approach produces more tenable saliency explanations compared to guided backpropagation and layer wise relevance propagation methods. The decoder part of the convolutional autoencoder can be used as a key that could be used to reconstruct the actual microstructure from encoded image information contributing to secure and efficient sharing of microstructure data. The proposed framework is generic and can be extended to identify important microstructural regions for other metals, composites, biomaterials, and material systems.

Efficient and Robust Method for Identifying Material Constant of Hyperelastic Material Model via Complex-step Derivative Approximation

An efficient and robust method for deriving the stress-strain relationship of hyperelastic material model (HM-model) was proposed via complex-step derivative approximation. This method has a capability to calculate the stress-strain relationship with the same accuracy as the analytical solution by coding only the strain energy density function, being applied for the implementation of an identification program of material constants for HM-model using a Microsoft/Excel. The performance evaluation of HM-models (Yeoh and Ogden models) using this method revealed that all models could reproduce the results of the three types of material tests (uniaxial tensile (UT), pure shear (PS) and equibiaxial tensile (BT) tests) of carbon-black-filled SBR. Moreover, only the hyperelastic model composed of the first invariant of the right Cauchy-Green tensor could predict the PS and BT test results from the UT data. As a future prospect, we would like to proceed with studies on other rubber materials and HM models.

Sensitivity analysis of system reliability using the complex-step derivative approximation

Sensitivity analysis of system failure probability is performed to evaluate the dependence of system reliability on design parameters or component events. Sensitivity analysis is integral for the implementation of efficient gradient-based optimization algorithms in system reliability-based design optimization, and for risk-informed decision-making. This paper presents a method for sensitivity analysis of system failure probability using complex-step differentiation. Many derivative approximations use a small step size to minimize subtractive cancellation errors. The complex-step approximation utilizes an imaginary number, such that the subtractive cancellation is not included in the formulation, resulting in calculations without the associated round-off error. The level of accuracy in sensitivity analysis using the finite difference method (FDM) can vary with change in the step size, which is generally selected arbitrarily, as the actual effect of the step size on the result itself is difficult to predict prior to actual calculation. The complex-step approximation, however, is not confined by the step size, as subtraction cancellation is not included. Compared to the FDM, the complex step approximation has only one limit, the numerical precision of evaluating the function. The proposed method integrates complex-step differentiation into a numerical integration scheme, for the assessment of system reliability and sensitivity. System failure probability and sensitivity are obtained by taking the real and imaginary part of the cumulative distribution function, which is numerically evaluated using the proposed method. The computational efficiency for system reliability problems involving high dimensionality is improved with the utilization of a dimension reduction technique. Numerical examples of sensitivity analysis for series, parallel, and general systems are presented to illustrate the performance of the proposed method.

Novel sensitivity method for evaluating the first derivative of the feed-forward neural network outputs

Evaluating the exact first derivative of a feedforward neural network (FFNN) output with respect to the input feature is pivotal for performing the sensitivity analysis of the trained neural network with respect to the inputs. In this paper, a novel method is presented that computes the analytical quality first derivative of a trained feedforward neural network output with respect to the input features without the need for backpropagation. To this end, the complex step derivative approximation is illustrated, and its implementation in the framework of the feedforward neural network is described. Artificial datasets are generated, and the efficacy of the proposed method for both regression and classification tasks is demonstrated. The results obtained for the regression task indicated that the proposed method is capable of obtaining analytical quality derivatives, and in the case of the classification task, the least relevant features could be identified.

Reliability-Based Design Optimization of Structures Using the Second-Order Reliability Method and Complex-Step Derivative Approximation

This paper proposes a reliability-based design optimization (RBDO) approach that adopts the second-order reliability method (SORM) and complex-step (CS) derivative approximation. The failure probabilities are estimated using the SORM, with Breitung’s formula and the technique established by Hohenbichler and Rackwitz, and their sensitivities are analytically derived. The CS derivative approximation is used to perform the sensitivity analysis based on derivations. Given that an imaginary number is used as a step size to compute the first derivative in the CS derivative method, the calculation stability and accuracy are enhanced with elimination of the subtractive cancellation error, which is commonly encountered when using the traditional finite difference method. The proposed approach unifies the CS approximation and SORM to enhance the estimation of the probability and its sensitivity. The sensitivity analysis facilitates the use of gradient-based optimization algorithms in the RBDO framework. The proposed RBDO/CS–SORM method is tested on structural optimization problems with a range of statistical variations. The results demonstrate that the performance can be enhanced while satisfying precisely probabilistic constraints, thereby increasing the efficiency and efficacy of the optimal design identification. The numerical optimization results obtained using different optimization approaches are compared to validate this enhancement.

The Complex-Step Derivative Approximation on Matrix Lie Groups

The complex-step derivative approximation is a numerical differentiation technique that can achieve analytical accuracy, to machine precision, with a single function evaluation. In this letter, the complex-step derivative approximation is extended to be compatible with elements of matrix Lie groups. As with the standard complex-step derivative, the method is still able to achieve analytical accuracy, up to machine precision, with a single function evaluation. Compared to a central-difference scheme, the proposed complex-step approach is shown to have superior accuracy. The approach is applied to two different pose estimation problems, and is able to recover the same results as an analytical method when available.

Modeling Zika Virus Transmission Dynamics: Parameter Estimates, Disease Characteristics, and Prevention

Because of limited data, much remains uncertain about parameters related to transmission dynamics of Zika virus (ZIKV). Estimating a large number of parameters from the limited information in data may not provide useful knowledge about the ZIKV. Here, we developed a method that utilizes a mathematical model of ZIKV dynamics and the complex-step derivative approximation technique to identify parameters that can be estimated from the available data. Applying our method to epidemic data from the ZIKV outbreaks in French Polynesia and Yap Island, we identified the parameters that can be estimated from these island data. Our results suggest that the parameters that can be estimated from a given data set, as well as the estimated values of those parameters, vary from Island to Island. Our method allowed us to estimate some ZIKV-related parameters with reasonable confidence intervals. We also computed the basic reproduction number to be from 2.03 to 3.20 across islands. Furthermore, using our model, we evaluated potential prevention strategies and found that peak prevalence can be reduced to nearly 10% by reducing mosquito-to-human contact by at least 60% or increasing mosquito death by at least a factor of three of the base case. With these preventions, the final outbreak-size is predicted to be negligible, thereby successfully controlling ZIKV epidemics.

Computation of High-Order Electromagnetic Field Sensitivities With FDTD and the Complex-Step Derivative Approximation

This paper introduces a new approach for the computation of electromagnetic field derivatives, up to any order, with respect to the material and geometric parameters of a given geometry, in a single Finite-Difference Time-Domain (FDTD) simulation. The proposed method is based on embedding the complex-step derivative (CSD) approximation into the standard FDTD update equations. Being finite-difference free, CSD provides accurate derivative approximations even for very small perturbations of the design parameters, unlike finite-difference approximations that are prone to subtractive cancellation errors. The availability of accurate approximations of field derivatives with respect to design parameters enables studies such as sensitivity analysis of multiple objective functions (as derivatives of those can be derived from field derivatives via the chain rule), uncertainty quantification, as well as multi-parametric modeling and optimization of electromagnetic structures. The theory, FDTD implementation and applications of this technique are presented.

Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models

Poro-elastic systems have been used extensively in modeling fluid flow in porous media in petroleum and earthquake engineering. Nowadays, they are frequently used to model fluid flow through biological tissues, cartilages, and bones. In these biological applications, the fluid-solid mixture problems, which may also incorporate structural viscosity, are considered on bounded domains with appropriate non-homogeneous boundary conditions. The recent work in [ 12 ] provided a theoretical and numerical analysis of nonlinear poro-elastic and poro-viscoelastic models on bounded domains with mixed boundary conditions, focusing on the role of visco-elasticity in the material. Their results show that higher time regularity of the sources is needed to guarantee bounded solution when visco-elasticity is not present. Inspired by their results, we have recently performed local sensitivity analysis on the solutions of these fluid-solid mixture problems with respect to the boundary source of traction associated with the elastic structure [ 3 ]. Our results show that the solution is more sensitive to boundary datum in the purely elastic case than when visco-elasticity is present in the solid matrix. In this article, we further extend this work in order to include local sensitivities of the solution of the coupled system to the boundary conditions imposed on the Darcy velocity. Sensitivity analysis is the first step in identifying important parameters to control or use as control terms in these poro-elastic and poro-visco-elastic models, which is our ultimate goal.

On the accuracy of the Complex-Step-Finite-Difference method

The Complex-Step-Finite-Difference method (CSFDM) is a very simple methodology that can be implemented in well known numerical techniques helping to improve, for instance in the wave propagation problem, time and/or space derivative based wavefields. We clarify differences between the CSFDM and previous implementations of the Complex-Step (CS) derivative approximation in well known numerical techniques. We study dispersion properties for the one-way and two-way wave equations using the Finite-Difference method (FDM), the Pseudospectral method (PSM), the Finite-Element method (FEM) and the CSFDM, under the influence of a plane wave and Ricker source time functions. We show the gain in numerical accuracy offered by the methodology of the CSFDM over the FDM, PSM and FEM. We finally discuss consequences of the CSFDM in future scenarios and propose directions of study in this area.

Complex-step derivative approximation in noisy environment

The complex-step derivative approximation is a powerful method for derivative approximations which has been successfully implemented in deterministic numerical algorithms. We explore and analyze its implementation in noisy environment through examples, error analysis and application to optimization methods. Numerical results show a promising performance of the complex-step gradient approximation in noisy environment.

Sensitivity via the complex-step method for delay differential equations with non-smooth initial data

In this report, we use the complex-step derivative approximation technique to compute sensitivities for delay differential equations (DDEs) with non-smooth (discontinuous and even distributional) history functions. We compare the results with exact derivatives and with those computed using the classical sensitivity equations whenever possible. Our results demonstrate that the implementation of the complex-step method using the method of steps and the Matlab solver dde23 provides a very good approximation of sensitivities as long as discontinuities in the initial data do not cause loss of smoothness in the solution: that is, even when the underlying smoothness with respect to the initial data for the Cauchy-Riemann derivation of the method does not hold. We conclude with remarks on our findings regarding the complex-step method for computing sensitivities for simpler ordinary differential equation systems in the event of lack of smoothness with respect to parameters.

Determination of the tangent stiffness tensor in materials modeling in case of large deformations by calculation of a directed strain perturbation

The Finite Element Method in the field of materials modeling is often relying to the tangent stiffness tensor of the constitutive law. This so called Jacobian matrix is required at each time increment and describes the local material behavior. It assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state. For increasingly sophisticated material models the tangent stiffness tensor can be derived analytically only with great effort, if at all. Numerical methods like the forward-difference, the central-difference, and the complex-step derivative approximation approach are widely used for its calculation. For each of these methods it is necessary to generate a specific strain perturbation. However, in large strain formulations it is not possible to perturb the strain directly but one can only modify the deformation gradient.We present our methods to generate a directed strain perturbation for the Green–Lagrange, Euler–Almansi and logarithmic strain measures as a function of the deformation gradient and compare them with other commonly used methods. An increase in accuracy and rate of convergence can be achieved with the proposed procedures.

Complex step derivative approximation of consistent tangent operators for viscoelasticity based on fractional calculus

In this contribution, the convergence behaviour of simulations with nonlinear viscoelastic models for rubber-like materials using fractional derivatives is investigated. Based on the complex step d...