Matti Schneider

• Dr. rer. nat.
• Professor at University of Duisburg-Essen

Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution...

Imposing nonperiodic boundary conditions for unit cell analyses may be necessary for a number of reasons in applications, for example, for validation purposes and specific computational setups. The work at hand discusses a strategy for utilizing the powerful technology behind fast Fourier transform (FFT)-based computational micromechanics—initially...

We discuss how Dirichlet boundary conditions can be directly imposed for the Moulinec–Suquet discretization on the boundary of rectangular domains in iterative schemes based on the fast Fourier transform (FFT) and computational homogenization problems in mechanics. Classically, computational homogenization methods based on the fast Fourier transfor...

The computational efficiency of FFT‐based computational micromechanics is deeply rooted in the underlying regular, that is, Cartesian, discretization. The bottleneck for most industrial applications is evaluating the typically rather expensive constitutive law on the regular grid. In the work at hand, we exploit coarsening strategies to evaluate th...

We introduce an approach to computational homogenization which unites the accuracy of interface‐conforming finite elements (FEs) and the computational efficiency of methods based on the fast Fourier transform (FFT) for two‐dimensional thermal conductivity problems. FFT‐based computational homogenization methods have been shown to solve multiscale p...

We extend the laminate based framework of direct deep material networks (DMNs) to treat suspensions of rigid fibers in a non‐Newtonian solvent. To do so, we derive two‐phase homogenization blocks that are capable of treating incompressible fluid phases and infinite material contrast. In particular, we leverage existing results for linear elastic la...

We present a fused sequential addition and migration (fSAM) algorithm to generate synthetic microstructures for long fiber reinforced hybrid composites. To incorporate the bending of long fibers, we model the fibers as polygonal chains and use an optimization framework to search for a non-overlapping fiber configuration with the desired properties....

Porous microstructures represent a challenge for the convergence of FFT‐based computational homogenization methods. In this contribution, we show that the damped Eyre–Milton iteration is linearly convergent for a class of nonlinear composites with a regular set of pores, provided the damping factor is chosen between zero and unity. First, we show t...

Computational homogenization may be used to evaluate effective properties of microstructured materials. As digital imaging techniques generally generate microstructure information on a regular grid, a common approach to compute the effective properties of the microstructured materials is to use fast Fourier transform (FFT)‐based methods. Traditiona...

This paper investigates the computational homogenization of thermal conductivity problems using a finite Radon transform as proposed by Derraz and coworkers, implemented using the Fourier slice theorem to allow utilization of the fast Fourier transform‐based frameworks and methods. For the finite Radon transform both the original approach and the c...

We present efficient algorithms for computing fourth‐order fiber orientation tensors associated to the exact closure. Based on quadrature using spherical designs, we investigate Newton's method as well as a fixed point scheme. These methods serve as alternatives to the original approach utilizing elliptic integrals. We compare the convergence behav...

Computational homogenization methods reveal the physical behavior of heterogeneous materials, often complex due to intricate manufacturing. Micromechanical techniques use microstructural data to predict effective behavior. However, processing digital material data, represented as voxels, involves handling numerous image points. While modern FFT-bas...

Computational homogenization methods offer deeper insights into the mechanical behavior of heterogeneous materials, often exhibiting complex anisotropic properties due to intricate manufacturing processes. Micromechanical techniques exploit the knowledge of the material's microstructure to predict effective behavior. Digital material data, represen...

This work explores connections between FFT-based computational micromechanics and a homogenization approach based on the finite Radon transform introduced by Derraz and co-workers. We revisit periodic homogenization from a Radon point of view and derive the multidimensional Radon series representation of a periodic function from scratch. We introdu...

We introduce the fused sequential addition and migration (fSAM) algorithm for generating microstructures of fiber composites with long, flexible, nonoverlap- ping fibers and industrial volume fractions. The proposed algorithm is based on modeling the fibers as polygonal chains and enforcing, on the one hand, the nonoverlapping constraints by an opt...

We extend the laminate based framework of direct Deep Material Networks (DMNs) to treat suspensions of rigid fibers in a non-Newtonian solvent. To do so, we derive two-phase homogenization blocks that are capable of treating incompressible fluid phases and infinite material contrast. In particular, we leverage existing results for linear elastic la...

In this article, we combine a Fast Fourier Transform based computational approach and a supervised machine learning strategy to discover models for the anisotropic effective viscosity of shear-thinning fiber suspensions. Using the Fast Fourier Transform based computational approach, we study the effects of the fiber orientation state and the impose...

A key challenge for the virtual characterization of components manufactured using short fiber-reinforced thermoplastics (SFRTs) is the inherent anisotropy which stems from the manufacturing process. To address this, a multi-scale approach is necessary, leveraging deep material networks (DMNs) as a micromechanical surrogate, for a one-stop solution...

We describe an algorithm for generating fiber-filled volume elements for use in computational homogenization schemes which accounts for a coupling of the fiber-length and the fiber-orientation. For prescribed fiber-length distribution and fiber-orientation tensor of second order, a maximum-entropy estimate is used to produce a fiber-length-orientat...

This work deals with the composite voxel method, which—in its original form—furnishes voxels containing more than one material with a surrogate material law accounting for the heterogeneity in the voxel. We show that the laminate composite voxel technique naturally arises as an assumed strain method, that is, the general framework introduced by Sim...

To compute the effective properties of random heterogeneous materials, a number of different boundary conditions are used to define the apparent properties on cells of finite size. Typically, depending on the specific boundary condition, different numerical methods are used. The article at hand provides a unified framework for Lippmann–Schwinger so...

We establish a computational methodology to incorporate interfaces with lower crack energy than the surrounding phases when computing the effective crack energy of brittle composite materials. Recent homogenization results for free discontinuity problems are directly applicable to the time-discretized Francfort-Marigo model of brittle fracture in t...

Moulinec and Suquet introduced a method for computational homogenization based on the fast Fourier transform which turned out to be rather computationally efficient. The underlying discretization scheme was subsequently identified as an approach based on trigonometric polynomials, coupled to the trapezoidal rule to substitute full integration. For...

Phase‐field models for the quasi‐static simulation of brittle fracture where the crack is approximated by a damage phase‐field are limited by the necessary memory and computation time. In this contribution, we study the applicability of low‐rank methods to phase‐field fracture models, specifically the tensor train (TT) format. To this end, we inves...

In short‐fiber reinforced polymers, fatigue damage is typically characterized by measuring the dynamic stiffness and its degradation under cyclic loading. Computational homogenization methods may be used to characterize the fatigue behavior of the composite via numerical predictions. Such an approach may reduce the experimental effort significantly...

Deep material networks (DMN) are a data‐driven homogenization approach that show great promise for accelerating concurrent two‐scale simulations. As a salient feature, DMNs are solely identified by linear elastic precomputations on representative volume elements. After parameter identification, DMNs act as surrogates for full‐field simulations of s...

Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation...

We present an adaption of the Orientation Corrected Shaking (OCS) method, originally developed for discontinuous short fiber-reinforced polymers, for generating continuous discontinuous fiberreinforced polymers. Due to the combination of continuous and discontinuous reinforced phases, the interface between the two layers needs to be accounted for i...

In this work, we propose to use deep material networks (DMNs) as a surrogate model for full-field computational homogenization to inversely identify material parameters of constitutive inelastic models for short fiber-reinforced thermoplastics (SFRTs). Micromechanics offers an elegant way to obtain constitutive models of materials with complex micr...

Under fatigue loading, the stiffness decrease in short-fiber reinforced polymers reflects the gradual degradation of the material. Thus, both measuring and modeling this stiffness is critical to investigate and understand the entire fatigue process. Besides evolving damage, viscoelastic effects within the polymer influence the measured dynamic stif...

Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation...

The manufacturing process of Sheet Molding Compound (SMC) influences the properties of a component in a non-deterministic fashion. To predict this influence on the mechanical performance, we develop a virtual process chain acting as a digital twin for SMC specimens from compounding to failure. More specifically, we inform a structural simulation wi...

Recent stochastic homogenization results for the Francfort–Marigo model of brittle fracture under anti-plane shear indicate the existence of a representative volume element. This homogenization result includes a cell formula which relies on Dirichlet boundary conditions. For other material classes, the boundary conditions do not effect the effectiv...

We provide theoretical investigations and empirical evidence that the effective stresses in computational homogenization of inelastic materials converge with a higher rate than the local solution fields. Due to the complexity of industrial‐scale microstructures, computational homogenization methods often utilize a rather crude approximation of the...

The manufacturing process of Sheet Molding Compound (SMC) influences the properties of a component in a non-deterministic fashion. To predict this influence on the mechanical performance, we develop a virtual process chain acting as a digital twin for SMC specimens from compounding to failure. More specifically, we inform a structural simulation wi...

This work is concerned with synthetic microstructure models of polycrystalline materials. Once a representation of the microstructure is generated, the individual grains need to be furnished with suitable crystal orientations, matching a specific crystal orientation distribution. We introduce a novel method for this task, which permits to prescribe...

The power of FFT‐based methods in computational micromechanics critically depends on a seamless integration of discretization scheme and solution method. In contrast to solution methods, where options are available that are fast, robust and memory‐efficient at the same time, choosing the underlying discretization scheme still requires the user to m...

We describe a Sequential Addition and Migration (SAM) algorithm for generating microstructures of fiber‐reinforced composites with a direct control of the magnitude of curvature of the fibers. The algorithm permits to generate microstructures with fibers that are significantly longer than the edge lengths of the underlying cell. Industrially proces...

For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration and enable...

Deep material networks (DMNs) are a recent multiscale technology which enable running concurrent multiscale simulations on industrial scale with the help of powerful surrogate models for the micromechanical problem. Classically, the parameters of the DMNs are identified based on linear elastic precomputations. Once the parameters are identified, DM...

We describe an algorithm for generating fiber-filled volume elements for use in computational homogenization schemes. The algorithm permits to prescribe both a length distribution and a fiber-orientation tensor of second order, and composites with industrial filler fraction can be generated. Typically, for short-fiber composites, data on the fiber-...

For effective cutting tool inserts that absorb thermal shock at varying temperature gradients, improved thermal conductivity and toughness are required. In addition, parameters such as the coefficient of thermal expansion must be kept within a reasonable range. This work presents a novel material design framework based on a multi-scale modeling app...

With the complexity of modern microstructured materials, computational homogenization methods have been shown to provide accurate estimates of their effective mechanical properties, reducing the involved experimental effort considerably. After solving the balance of linear momentum on the microscale, the effective stress is traditionally computed t...

We extend the FE-DMN method to fully coupled thermomechanical two-scale simulations of composite materials. In particular, every Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN). Such a DMN serves as a high-fidelity surrogate model for full-field solutions on the microscopic scale of inelastic, non-...

Phase-field models permit accounting for the underlying physics of the microstructure evolution process when simulating the emergent microstructure of a variety of multiscale materials. As an inherent characteristic, the individual phases (or states) of the material are not known in advance and only emerge upon evolution. To account for this matter...

Short-fiber reinforced materials show material degradation under fatigue loading prior to failure. To investigate these effects, we model the constituents by an isotropic fatigue damage model for the matrix material and isotropic linear-elastic material model for the fibers. On the microscale we compute the overall material response for cell proble...

We present a holistic multiscale approach for constructing anisotropic criteria describing the macro- scopic failure of sheet molding compound composites based on full-field simulations of microscale damage evolution. We use an anisotropic damage model on the microscale that directly operates on the compliance tensor, captures matrix and bundle dam...

Characterizing short-fiber reinforced polymers under high-cycle fatigue loading is a tedious experimental task. To reduce the necessary experiments to a minimum, we introduce a computational strategy involving a mean-stress dependent fatigue-damage model for the stiffness degradation in short-fiber reinforced polymers. The key challenge in these ma...

Directionally solidified eutectics of NiAl matrix and fibrous refractory metals, like Mo, can form cellular mesostructures with significant fiber misalignment and changing fiber volume fraction, for example, when being solidified at high growth rates or when increased solidification intervals are present in the alloys. In order to reveal the deteri...

Under fatigue-loading, short-fiber reinforced thermoplastic materials typically show a progressive degradation of the stiffness tensor. The stiffness degradation prior to failure is of primary interest from an engineering perspective, as it determines when fatigue cracks nucleate. Efficient modeling of this fatigue stage allows the engineer to moni...

Deep material networks (DMN) are a promising piece of technology for accelerating concurrent multiscale simulations. DMNs are identified by linear elastic pre-computations on representative volume elements, and serve as high-fidelity surrogates for full-field simulations on microstructures with inelastic constituents. The offline training phase is...

Recently, mathematically well-defined homogenization results for the Francfort-Marigo fracture model were established. To solve the resulting cell formula, efficient computational methods were developed and improvements on solver and discretization techniques were investigated. We discuss an approach for solving the governing cell formula based on...

In this work, we propose a fully coupled multiscale strategy for components made from short fiber reinforced composites, where each Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN) which covers the different fiber orientation states varying within the component. These DMNs need to be identified by l...

We investigate volume-element sampling strategies for the stochastic homogenization of particle-reinforced composites and show, via computational experiments, that an improper treatment of particles intersecting the boundary of the computational cell may affect the accuracy of the computed effective properties. Motivated by recent results on a supe...

We extend the FE-DMN method to fully coupled thermomechanical two-scale simulations of composite materials. In particular, every Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN). Such a DMN serves as a high-fidelity surrogate model for full-field solutions on the microscopic scale of inelastic, non-...

A variety of materials, such as polycrystalline ceramics or carbon fiber reinforced polymers, show a pronounced anisotropy in their local crack resistance. We introduce an FFT-based method to compute the effective crack energy of heterogeneous, locally anisotropic materials. Recent theoretical works ensure the existence of representative volume ele...

In this work, we advocate using Bayesian techniques for inversely identifying material parameters for multiscale crystal plasticity models. Multiscale approaches for modeling polycrystalline materials may significantly reduce the effort necessary for characterizing such material models experimentally, in particular when a large number of cycles is...

Polarization-type methods are among the fastest solution methods for FFT-based computational micromechanics{. However, their performance depends critically on the choice of the reference material. Only for finitely contrasted materials, optimum-selection strategies are known. This work focuses on adaptive strategies for choosing the reference mater...

Understanding the effective viscosity of fiber-filled polymer melts is essential for predicting the local fiber orientation of injection molded short-fiber reinforced components. To circumvent the intrinsic difficulties of experimentally determining the strongly anisotropic viscosity of such particle-reinforced melts, an FFT-based computational hom...

This work is devoted to anisotropic continuum-damage mechanics in the quasi-static, isothermal, small-strain setting. We propose a framework for anisotropic damage evolution based on the compliance tensor as primary damage variable, in the context of generalized standard models for dissipative solids. Based on the observation that the Hookean strai...

This work is concerned with computing the of periodic and random media which arises in mathematical homogenization results for the Francfort‐Marigo model of brittle fracture. A previous solver based on the fast Fourier transform (FFT) led to solution fields with ringing or checkerboard artifacts and was limited in terms of the achievable accuracy....

In this work, we propose a fully coupled multiscale strategy for components made from short fiber reinforced composites, where each Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN) which covers the different fiber orientation states varying within the component. These DMNs need to be identified by l...

We investigate volume-element sampling strategies for the stochastic homogenization of particle-reinforced composites and show, via computational experiments, that an improper treatment of particles intersecting the boundary of the computational cell may affect the accuracy of the computed effective properties. Motivated by recent results on a supe...

We introduce an FFT-based solver for the combinatorial continuous maximum flow discretization applied to computing the minimum cut through heterogeneous microstructures. Recently, computational methods were introduced for computing the effective crack energy of periodic and random media. These were based on the continuous minimum cut-maximum flow d...

Experimental studies of Chebbi et al. [1] on fatigue loading of fiber‐reinforced polymers have shown that there is a phase of stable stiffness decrease prior to growing fatigue cracks. Modeling this stiffness degradation is an essential step in understanding fatigue effects of these materials. The constitutive behavior of short‐fiber reinforced pol...

Classical solution methods in FFT‐based computational micromechanics operate on, either, compatible strain fields or equilibrated stress fields. In contrast, polarization schemes are primal‐dual methods whose iterates are neither compatible nor equilibrated. Recently, it was demonstrated that polarization schemes may outperform the classical method...

In this work, we investigate a model for the anisotropic and loading-direction dependent stiffness degradation of short-fiber reinforced thermoplastics subjected to high-cycle fatigue loading. Based upon the variational setting of generalized standard materials, we model the stiffness degradation of the matrix in cycle space by a simple isotropic f...

Thermomechanical couplings are present in many materials and should therefore be considered in multiscale approaches. Specific cases of thermomechanical behavior are the isothermal and the adiabatic regime, in which the behavior of real materials differs. Based on the consistent asymptotic homogenization framework for thermomechanically coupled gen...

Ideas from the mathematical theory of optimal transport have recently been transferred to the micromechanics of polycrystalline materials, leading to fast methods for generating polycrystalline microstructures with grains of prescribed volume fraction in terms of centroidal Laguerre tessellations. In this work, we improve the state of the art solve...

For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration, and enabl...

For fast Fourier transform (FFT)-based computational micromechanics, solvers need to be fast, memory-efficient, and independent of tedious parameter calibration. In this work, we investigate the benefits of nonlinear conjugate gradient (CG) methods in the context of FFT-based computational micromechanics. Traditionally, nonlinear CG methods require...

We show that, under suitable hypotheses on the non‐porous material law and a geometric regularity condition on the pore space, Moulinec‐Suquet’s basic solution scheme converges linearly. We also discuss for which derived solvers a (super‐)linear convergence behavior may be obtained, and for which such results do not hold, in general. The key techni...

We study fast and memory-efficient FFT-based implicit solution methods for small-strain phase-field crack problems for microstructured brittle materials. A fully implicit first order formulation of the problem coupling elasticity and damage permits using comparatively few, but large, time steps compared to semi-explicit schemes. We investigate memo...

We investigate deep material networks (DMNs), recently introduced by Liu-Wu-Koishi [Comput. Method Appl. M., vol. 345, pp. 1138–1168, 2019], from the viewpoint of classical micromechanics at small strains. We aim to establish the basic micromechanical principles of deep material networks, shed light on the characteristics of the building blocks and...

Sheet molding compound (SMC) composites combine high lightweight potential with excellent formability and are frequently used in industrial applications. To reduce safety factors in dimensioning SMC parts, the influence of processing parameters and stochastic variation of microstructural and physical properties needs to be quantified accurately. Ta...

Computational homogenization schemes based on the fast Fourier transform (FFT) enable studying the effective micromechanical behavior of polycrystalline microstructures with complex morphology. In the conventional strain-based setting, evaluating the single crystal elasto-viscoplastic constitutive law involves solving a non-linear system of equatio...

This work is devoted to investigating the computational power of Quasi‐Newton methods in the context of fast Fourier transform (FFT)‐based computational micromechanics. We revisit (FFT)‐based Newton‐Krylov solvers as well as modern Quasi‐Newton approaches such as the recently introduced Anderson accelerated basic scheme. In this context, we propose...

Cell formulae for the effective crack resistance of a heterogeneous medium obeying Francfort‐Marigo's formulation of linear elastic fracture mechanics have been proved recently, both in the context of periodic and stochastic homogenization. This work proposes a numerical strategy for computing the effective, possibly anisotropic, crack resistance o...

We present a Lippmann‐Schwinger equation for the explicit jump discretization of thermal computational homogenization. Our solution scheme is based on the fast Fourier transform and thus fast and memory‐efficient. We reformulate the explicit jump discretization using harmonically averaged thermal conductivities and obtain a symmetric positive defin...

The description of material failure as an energy minimization problem, i.e., the Francfort–Marigo model, has been studied widely in recent years. The approximation of the crack surface as a phase field, i.e., smeared interface, enjoys great popularity, as it allows describing fracture as a set of partial differential equations. In numerical homogen...

We revisit the polarization-based schemes introduced to FFT-based computational homogenization by Eyre–Milton, Michel–Moulinec–Suquet and Monchiet–Bonnet. When applied to nonlinear problems, these polarization-based methods suffer from two handicaps. Firstly, the optimal choice of algorithmic parameters is only known for the linear elastic case. Se...

In this work, we compare measurements of strength, permeability and thermal conductivity of blown anorganically bound sand cores to simulation results on digitally generated unit cell models resolving the fine sand grain-binder aggregate on the micrometer scale. The quality and operationality of sand cores used in foundry applications is strongly a...

We investigate, both mathematically and numerically, the self-consistent clustering analysis recently introduced by Liu–Bessa–Liu and, independently, by Wulfinghoff–Cavaliere–Reese. We establish, in the small strain setting and non-softening material behavior, existence and uniqueness of the solution to the discretized equations for fixed (possibly...

The compression molding of sheet molding compounds (SMCs) is typically thought of as a fluid mechanics problem. The usage of CF-SMC with high fiber volume content (over 50%) and long fiber reinforcement structures (up to 50 mm) challenges the feasibility of this point of view. In this work a user-defined material model based on a solid mechanics fo...

To predict the nonlinear mechanical behavior of components made of short fiber-reinforced plastics (SFRP) under long term and cyclic loading, coupled process and component simulations are required. The injection molding process leads to locally varying fiber orientations within the component. This varying microstructure [1] significantly influences...

We present a variational formulation and a Lippmann‐Schwinger equation for the explicit jump discretization of thermal computational homogenization problems, together with fast and memory‐efficient matrix‐free solvers based on the fast Fourier transform. Wiegmann and Zemitis introduced the explicit jump discretization for volumetric image‐based com...

Building upon the equivalence of Moulinec‐Suquet's basic scheme with gradient descent methods, we investigate the effect of using the celebrated Barzilai‐Borwein step size selection technique in this context. We provide an overview of recent convergence theory and present efficient implementations in the context of computational micromechanics, wit...

For short fiber reinforced plastic parts the local fiber orientation has a strong influence on the mechanical properties. To enable multiscale computations using surrogate models we advocate a two-step identification strategy. Firstly, for a number of sample orientations an effective model is derived by numerical methods available in the literature...

In this article we model sand core materials on the micro-meter scale, resolving individual sand grains and binding bridges, to obtain effective elastic moduli of the composite by computational homogenization, laying the foundations for investigating the strength properties of core blown parts with foundry applications. We analyze sand core materia...

In this article we introduce a fiber orientation-adapted integration scheme for Tucker’s orientation averaging procedure applied to non-linear material laws, based on angular central Gaussian fiber orientation distributions. This method is stable w.r.t. fiber orientations degenerating into planar states and enables the construction of orthotropic h...

Deep drawing of paperboard with rigid tools and immediate compression has only a small presence in the market for secondary packaging solutions due to a lack of understanding of the physical relations that occur during the forming process. As with other processes that deal with interactions between two solids in contact, the control of the factors...

The composite voxel technique was developed in the framework of linear elasticity and hyperelasticity for regular voxel grid discretizations which cannot resolve material interfaces exactly in general. In this work, we study the inelastic behavior of two-phase laminates. In particular, we derive an analytical nonlinear formula for the unknown rank...

(Online: http://publica.fraunhofer.de/documents/N-438698.html) Fibre mats for medium-density fibreboards (MDF) are normally formed without dedicated fibre orientation. Simulations of the micro-structure of MDF based on real particle size distributions and micro-tomographic image data revealed that a horizontal fibre orientation results in an increa...

We present an algorithm for generating volume elements of short fiber reinforced plastic microstructures for prescribed fourth order fiber orientation tensor, fiber aspect ratio and solid volume fraction. The algorithm inserts fibers randomly into an existing microstructure, and removes the resulting overlap systematically based on a gradient desce...