Interface-enriched Generalized Finite Element Method Research Articles

Multiscale design of nonlinear materials using reduced-order modeling

A two-step optimization method is proposed for the efficient multiscale design of heterogeneous materials whose nonlinear macroscopic response is driven by volumetric and interfacial damage taking place at the microstructural level. The Eigendeformation-based reduced-order Homogenization Method (EHM) is used in a reduced-order design phase, allowing for a preliminary design that combines multiple initial configurations with a gradient-based optimization method. The preliminary design is then adopted to initialize a high-fidelity optimization phase based on the Interface-Enriched Generalized Finite Element Method (IGFEM) to arrive at the final design. The analytic sensitivities of EHM with respect to nonlinear material properties are derived to drive the gradient-based algorithm. Several multiscale optimization problems of increasing complexity are presented to illustrate the efficiency of the proposed method using an objective function based on stress error relative to a desired stress–strain response.

On tailoring fracture resistance of brittle structures: A level set interface-enriched topology optimization approach

We propose a fully immersed topology optimization procedure to design structures with tailored fracture resistance under linear elastic fracture mechanics assumptions for brittle materials. We use a level set function discretized by radial basis functions to represent the topology and the Interface-enriched Generalized Finite Element Method (IGFEM) to obtain an accurate structural response. The technique assumes that cracks can nucleate at right angles from the boundary, at the location of enriched nodes that are added to enhance the finite element approximation. Instead of performing multiple finite element analyses to evaluate the energy release rates (ERRs) of all potential cracks—a procedure that would be computationally intractable—we approximate them by means of topological derivatives after a single enriched finite element analysis of the uncracked domain. ERRs are then aggregated to construct the objective function, and the corresponding sensitivity formulation is derived analytically by means of an adjoint formulation. Several numerical examples demonstrate the technique’s ability to tailor fracture resistance, including the well-known benchmark L-shaped bracket and a multiple-loading optimization problem for obtaining a structure with fracture resistance anisotropy.

A GFEM-based reduced-order homogenization model for heterogeneous materials under volumetric and interfacial damage

This manuscript presents a multiscale reduced-order modeling framework for heterogeneous materials that accounts for both cohesive interface failure and continuum damage. The model builds on the eigendeformation-based reduced-order homogenization model (EHM), which relies on the pre-calculation of a set of coefficient tensors that account for the effects of linear and nonlinear material behavior between regions of the domain known as parts. A k-means clustering algorithm is used to optimally construct these parts and a new formulation for the partitioning of interfaces using this method is proposed. The extraction of the volumetric and interfacial influence functions is performed using the Interface-enriched Generalized Finite Element Method (IGFEM), which relies on a finite element discretization that does not conform to the material phase boundaries. A Lagrange multiplier method is used in this preprocessing phase, allowing for the reuse of the matrix factorization for different influence function problems and hence leading to efficiency improvement. A newly proposed traction calculation for the interface partition is also adopted to alleviate the instability caused by traction calculations along interfaces. The accuracy and efficiency of the IGFEM–EHM method is assessed through comparison with reference IGFEM simulations. The method is then used to extract the nonlinear multiscale response of particulate, unidirectional fiber-reinforced, and woven composites.

An interface-enriched generalized finite element method for level set-based topology optimization

During design optimization, a smooth description of the geometry is important, especially for problems that are sensitive to the way interfaces are resolved, e.g., wave propagation or fluid-structure interaction. A level set description of the boundary, when combined with an enriched finite element formulation, offers a smoother description of the design than traditional density-based methods. However, existing enriched methods have drawbacks, including ill-conditioning and difficulties in prescribing essential boundary conditions. In this work, we introduce a new enriched topology optimization methodology that overcomes the aforementioned drawbacks; boundaries are resolved accurately by means of the Interface-enriched Generalized Finite Element Method (IGFEM), coupled to a level set function constructed by radial basis functions. The enriched method used in this new approach to topology optimization has the same level of accuracy in the analysis as the standard finite element method with matching meshes, but without the need for remeshing. We derive the analytical sensitivities and we discuss the behavior of the optimization process in detail. We establish that IGFEM-based level set topology optimization generates correct topologies for well-known compliance minimization problems.

How to design a blockage-tolerant cooling network?

Nature-inspired microvascular composites rely on microchannel networks to enable a variety of multifunctionality including self-healing and active cooling. However, the small size of the microchannels makes them highly susceptible to blockage from the presence of particulates in the fluid or damage in the vascular network. Blockage in the microchannels may disrupt the functionality of the microvascular composites. As an effort to alleviate the destructive effects of microchannel blockage, we present a Hybrid Topology/Shape (HyTopS) optimization approach to design actively-cooled microvascular networks for blockage tolerance. The proposed method shows a promising capability in producing a reliable design for blockage tolerance. In this hybrid scheme, the optimizer can modify the topology of the design during the shape optimization procedure. Being able to change the topology of the network enables the optimizer to provide network redundancy to effectively optimize the design for blockage tolerance. In this scheme, the optimizer takes into consideration the blockage scenarios in each iteration. As a result, the design will be optimized in such a way that it would perform satisfactorily when blockage of microchannels occurs. The main challenge in this method, similar to the other common approaches for blockage tolerance, is the extensive computational cost. To address this issue, we present a novel Importance Factor (IF) method and combined it with the HyTopS approach to reduce the computational cost significantly. For further reduction of computational cost, we use Interface-enriched Generalized Finite Element Method (IGFEM) wherein the design geometry is projected onto a fixed mesh. Thus, there is no need for remeshing during the optimization procedure. Moreover, a fully analytic sensitivity analysis of the shape optimization of the microvascular composites is also developed. Finally, we solve several numerical examples to investigate the pros and cons of using the proposed optimization approaches for blockage tolerance.

On the usefulness of gradient information in surrogate modeling: Application to uncertainty propagation in composite material models

In this work, the performance of non-gradient as well as gradient-enhanced versions of two different classes of surrogate modeling approaches, polynomial least squares regression and kernel based radial basis function interpolation, are compared in the context of a composite mechanics problem. Sequential space filling random designs are used for selecting the training points. The primary goal is to investigate whether additional gradient information obtained at a relatively small cost helps in generating surrogates of better quality compared to those obtained without any gradient information. It is found from the study that if the gradient and/or function evaluations are noisy, then the quality of the surrogate approximation is similar for both the gradient enhanced and the non-gradient based surrogate models. However, if the gradient and function evaluations are accurate, the gradient-enhanced surrogate models consistently perform better than the non-gradient based surrogate models, indicating that the gradient information enhances the quality of the surrogates. Low dimensional analytical test functions are used to demonstrate this behavior. As an application problem, we consider a multi-fiber reinforced composite model with a different interfacial damage parameter assigned to each fiber∕matrix interface. In particular, the surrogate describes the variation of the homogenized stress at a given input strain as a function of the interface damage parameters. The Interface-Enriched Generalized Finite Element Method (IGFEM) is used in this case to solve for the stress as well as the gradients of the stress with respect to the damage parameters. Thus the goal of this study is two-fold: (1) to compare the error convergence properties in surrogate modeling using different sequential random space filled designs, with and without gradient information; (2) to identify the circumstances in which additional gradient information is beneficial for surrogate modeling.

Gradient-based hybrid topology/shape optimization of bioinspired microvascular composites

Construction of bioinspired vasculature in synthetic materials enables multi-functional performance via mass transport through internal fluidic networks. However, exact reproduction of intricate, natural microvascular architectures is nearly impossible and thus there is a need to create practical, manufacturable designs guided by multi-physics principles. Here we present a Hybrid Topology/Shape (HyTopS) optimization scheme for microvascular materials using the Interface-enriched Generalized Finite Element Method (IGFEM). This new approach, which can simultaneously perform topological changes as well as shape optimization of microvascular materials, is demonstrated in the context of thermal regulation. In the current study, we present a new feature that enables the optimizer to augment network topology by creating/removing microchannels during the shape optimization process. This task has been accomplished by introducing a new set of design parameters, which act analogous to the penalization factor in the Solid Isotropic Material with Penalization (SIMP) method. The analytical sensitivity for the HyTopS optimization scheme has been derived and the sensitivity accuracy is verified against the finite difference method. We impose a set of geometrical constraints to account for manufacturing limitations and produce a design which is suitable for large-scale production without the need to perform post-processing on the obtained optimum. The method is validated by active-cooling experiments on vascularized carbon-fiber composites. Finally, we compare various application examples to demonstrate the advantages of the newly introduced HyTopS optimization scheme over solely shape optimization for microvascular materials.

IGFEM-based shape sensitivity analysis of the transverse failure of a composite laminate

This manuscript presents a shape sensitivity analysis method based on an Interface-enriched Generalized Finite Element Method (IGFEM) formulation and its application to the sensitivity of the transverse failure of a fiber-reinforced composite laminate with respect to the geometrical parameters that define its microstructure. The analytical sensitivities with respect to individual fiber radius and placement are first derived within the context of a cohesive IGFEM solver specially developed to simulate the fiber/matrix debonding observed in the transverse failure of composite laminates with high fiber volume fraction. The IGFEM solver utilizes $$C^{-1}$$ continuous enrichment functions and a cohesive failure model to capture the transverse cracking associated primarily with fiber/matrix interface debonding. In addition to the sensitivities with respect to individual geometrical parameters such as the radius of individual fibers, the sensitivities of the transverse stress–strain response with respect to the parameters that define the distributions of the geometrical parameters such as the mean and standard deviation of the fiber radius and nearest-neighbor distance distributions are also derived. The sensitivity analysis is performed on realistic microstructures composed of hundreds of fibers to characterize the influence of the geometrical parameters and their distributions on the transverse failure response of the composite laminate.

A gradient plasticity creep model accounting for slip transfer/activation at interfaces evaluated for the intermetallic NiAl-9Mo

Interfaces can act as dislocation obstacles, sinks or dislocation sources and, therefore, influence strongly the mechanical properties of metals. To consider these effects at high temperature creep, a three-dimensional gradient crystal plasticity model is introduced. The interaction between dislocations and the fiber-matrix interface is included by an interface flow rule, which accounts for the gradient stresses and the normal component of the Cauchy stress on the interface. Motivated by the interface-enriched generalized finite element method (IGFEM), continuous shape functions allowing for weak discontinuities are introduced. These shape functions are used to evaluate the interface flow rule at sharp interfaces and are validated by comparing numerical simulation results of a laminate for single slip with an analytical solution. To investigate the slip transfer/activation at the interface, the directionally solidified NiAl-9Mo composite is modeled as regular fibrous microstructure. The simulated creep curves agree well with experimentally measured ones. It is found that the stress dependency of the interface flow rule is necessary to reproduce the well known composite's Norton behavior. The simulations reveal that the creep behavior of the composite is mainly controlled by the fibers and the interface properties. Finally, the specific shape of the creep curve could be explained.

Design of redundant microvascular cooling networks for blockage tolerance

Microvascular networks can provide host materials with many functions including self-healing and active cooling. However, vascular networks are susceptible to blockage which can dramatically reduce their functional performance. A novel optimization scheme is presented to design networks that provide sufficient cooling capacity even when partially blocked. Microvascular polydimethylsiloxane (PDMS) panels subject to a 2000 W m−2 applied heat flux and 28.2 mL min−1 coolant flow rate are simulated using dimensionally reduced thermal and hydraulic models and an interface-enriched generalized finite element method (IGFEM). Channel networks are optimized to minimize panel temperature while the channels are either clear (the O0 scheme), subject to the single worst-case blockage (O1), or subject to two worst-case blockages (O2). Designs are optimized with nodal degree (a measure of redundancy) ranging from 2 to 6. The results show that blockage tolerance is greatly enhanced for panels optimized while considering blockages and for panels with higher nodal degree. For example, the 6-degree O1 design only has a temperature rise of 7 °C when a single channel is blocked, compared to a 35 °C rise for the 2-degree O0 design. Thermography experiments on PDMS panels validate the IGFEM solver and the blockage tolerance of optimized panels.

3D dimensionally reduced modeling and gradient-based optimization of microchannel cooling networks

This paper presents a dimensionally reduced thermal model and gradient-based shape optimization scheme for the 3D computational design of actively cooled panels. A correction method previously used in wire-based electromagnetics is applied to address convergence issues associated with the singularity of the thermal solution along the microchannels. The numerical solution is obtained with the interface-enriched generalized finite element method (IGFEM), which greatly simplifies mesh generation by allowing for the use of a non-conforming mesh to capture the temperature gradient discontinuity across the microchannels. The temperature distribution calculated with the IGFEM on coarse meshes agrees with that obtained using significantly more complex and costly ANSYS FLUENT simulations. We then combine the IGFEM with a sensitivity analysis and the sequential quadratic programming algorithm in MATLAB to solve two sets of shape optimization problems related to actively cooled microvascular composite panels. These problems demonstrate a key advantage of the IGFEM in avoiding severe mesh distortion during shape optimization. Lastly, we present a semi-analytical model based on the concept of the zone of influence of a channel to estimate the maximum temperature of an actively cooled plate with straight embedded microchannels.

Gradient-based design of actively-cooled microvascular composite panels

Recent advances in manufacturing based on sacrificial fiber or template techniques have allowed complex networks of microchannels to be embedded in microvascular composites. In the thermal application of interest, a novel battery packaging scheme for electric vehicles is considered where each battery is surrounded by microvascular composite panels for temperature regulation and structural protection. We use simplified thermal and hydraulics models validated against more complex 3D FLUENT simulations and experiments to obtain the surface temperature distribution of the panel and the pressure drops across the microchannels. We further eliminate the cost and complexity associated with mesh generation by applying the interface-enriched generalized finite element method (IGFEM), which allows a non-conforming mesh to capture the discontinuous temperature gradient across the microchannels. The IGFEM thermal solver is then combined with a gradient-based shape optimization scheme to obtain optimal designs of a set of branched microchannel networks. The design parameters are the channel control points, which define the shape of the network. We use the p-mean as a differentiable objective function in place of the maximum temperature. To obtain accurate gradients with respect to the design parameters efficiently, we perform a sensitivity analysis based on a recently developed adjoint method for IGFEM. Starting from many distinct configurations, we obtain the optimal designs for a wide range of network topologies. We also investigate the effect of the coolant flow rate on the optimal design.

A 3-D Interface-Enriched Generalized FEM for Electromagnetic Problems With Nonconformal Discretizations

An interface-enriched generalized finite-element method (IGFEM) is introduced for efficient 3-D electromagnetic analysis of heterogeneous materials. Without using meshes that conform to the material microstructures, which greatly lessens the burden of mesh generation, the method assigns generalized degrees of freedom (DOFs) at material interfaces to capture the discontinuities of the field and its derivatives. The generalized DOFs are supported by enriched vector basis functions (VBFs), which are constructed through a linear combination of the VBFs from the subelements. Several verification examples are provided to show that the IGFEM is not sensitive to the quality of the subelements and maintains the same level of solution accuracy and computational complexity as the standard finite-element method (FEM) based on conformal meshes. The potential of the proposed IGFEM is demonstrated by simulating some engineering problems with complex, periodic internal structures, including composite materials with randomly distributed spherical particles and ellipsoidal inclusions and microvascular channels.

A gradient-based shape optimization scheme using an interface-enriched generalized FEM

Abstract A gradient-based shape optimization scheme using an Interface-enriched Generalized Finite Element Method (IGFEM) is presented wherein the design geometry is projected onto a fixed mesh and the IGFEM is used for analysis. This approach eliminates the mesh distortion present in conventional Lagrangian shape optimization methods, as well as the need for remeshing. An analytical sensitivity analysis using both the adjoint or direct approaches is presented to compute derivatives of the objective and constraint functions. Due to the fixed nature of the mesh, the so-called design velocity field only needs to be computed on the structure boundary/interface. A comparison between IGFEM- and conventional FEM-based shape optimization schemes is presented, showing an improved precision for the IGFEM approach. Finally, we solve various numerical examples to demonstrate the capability of the method including the computational design of particulate and microvascular composites.

A NURBS-based interface-enriched generalized finite element scheme for the thermal analysis and design of microvascular composites

Motivated by recent advances in manufacturing techniques for high-temperature microvascular composites, a NURBS-based interface-enriched generalized finite element method (IGFEM) is developed to solve a simplified thermal model of microchannels embedded in the materials. This method is capable of handling curved and branched microchannels. Solutions more accurate than those achieved with the conventional finite element method can be obtained with coarse meshes that do not conform to the geometry of the microchannels. Near-optimal asymptotic convergence rate is also achieved with this method even for highly curved microchannels. The capability of the numerical scheme is demonstrated by solving problems with complex microchannel configurations.

Computational analysis of actively-cooled 3D woven microvascular composites using a stabilized interface-enriched generalized finite element method

The computational design of an actively-cooled 3D woven microvascular composite plate with sinusoidal and straight microchannels is presented. The design objectives include minimizing the maximum temperature of the composite, the microchannel volume fraction, and the pressure drop needed to circulate the coolant in the microchannels. We study the impact of a variety of parameters on the optimal design of a microvascular composite plate subjected to a uniform heat flux over its bottom surface. These parameters include the spacing, wavelength, and amplitude of the microchannels, the coolant type and flow rate, and the applied thermal loads. To facilitate the computational design process, a mesh-independent Interface-enriched Generalized Finite Element Method (IGFEM) is employed to evaluate the temperature field in the actively-cooled composite. The IGFEM solver also includes the streamline upwind Petrov-Galekin stabilization scheme to eliminate the spurious oscillations in the temperature field due to the convection-dominated heat transfer in the microchannels. This study reveals that the straight microchannels are often the optimal configuration. Design maps are presented to evaluate the required flow rate as a function of the applied thermal load and the plate dimensions.

Computational modeling and design of actively-cooled microvascular materials

The computational modeling and design of an actively-cooled microvascular fin specimen is presented. The design study is based on three objective functions: (i) minimizing the maximum temperature in the thermally loaded fin, (ii) optimizing the flow efficiency of the embedded microchannel, and (iii) minimizing the void volume fraction of the microvascular material. A recently introduced Interface-enriched Generalized Finite Element Method (IGFEM) is employed to evaluate the temperature field in a 2D model of the specimen, allowing for the accurate and efficient capturing of the gradient discontinuity along the fluid/solid interface without the need of meshes that conform to the geometry of the problem. Finding the optimal shape of the embedded microchannel is thus accomplished with a single non-conforming mesh for all configurations. Prior to the optimization study, the IGFEM solver is validated through comparison with infrared measurements of the thermal response of an epoxy fin with a sinusoidal microchannel.