Numerical analysis of anisotropic ductile continuum damage

The paper deals with the numerical analysis of large elastic–plastic deformation behavior of anisotropically damaged ductile solids based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics. Estimates of the stress and strain histories are obtained from a straightforward numerical integration algorithm based on operator split methodology which employs an inelastic (damage–plastic) predictor followed by an elastic corrector step. The finite element method is used to approximate the linearized variational problem. Furthermore, identification of material parameters is discussed. Numerical simulation of the elastic–plastic deformation behavior of damaged tension specimens demonstrate the efficiency of the formulation.     

A new mixed finite element based on different approximations of the minors of deformation tensors

Finite element formulations for arbitrary hyperelastic strain energy functions that are characterized by a locking-free behavior for incompressible materials, a good bending performance and accurate solutions for coarse meshes need still attention. Therefore, the main goal of this contribution is to provide an improved mixed finite element for quasi-incompressible finite elasticity. Based on the knowledge that the minors of the deformation gradient play a major role for the transformation of infinitesimal line-, area- and volume elements, as well as in the formulation of polyconvex strain energy functions a mixed finite element with different interpolation orders of the terms related to the minors is developed. Due to the formulation it is possible to condensate the mixed element formulation at element level to a pure displacement form. Examples show the performance and robustness of the element.     

Coupled continuum damage mechanics and crystal plasticity model and its application in damage evolution in polycrystalline aggregates

In the present study, damage initiation and growth in a polycrystalline aggregate are investigated. In this regard, an anisotropic continuum damage mechanics coupled with rate-dependent crystal plasticity theory is employed. Using a thermodynamically consistent procedure, a finite deformation formulation is derived. For this purpose, the damage tensor is incorporated in the crystal plasticity formulation for a cubic single crystal. The damage evolution is considered to be dependent on the history of damage, equivalent plastic strain, stress triaxiality, and Lode parameters. This material model is implemented in the commercial finite-element code Abaqus/Standard by developing a user material subroutine (UMAT). Using the available experimental tests of 316L single crystal in the literature, the crystal plasticity hardening and damage parameters are calibrated considering the stress–strain curve before and after necking, respectively. The damage sites in a single-phase polycrystalline aggregate are also considered using a polycrystalline model consisting of grains with random sizes and orientations. The results show that the damage arises at the grain boundaries and triple junctions. Moreover, growth of the damage mostly occurs in the grains with higher Schmid factor compared to the neighboring grains. The presented model manifests capacity for determination of damage initiation sites and damage evolution in polycrystalline models.

     

Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals

We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with the wavelength of light. Depending on the geometry, material properties and lattice structure these materials exhibit a photonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal. By Bloch/Floquet theory, this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions, which is discretized with a mixed discontinuous Galerkin (DG) formulation using modified Nédélec basis functions. We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation. To guarantee the non-pollution of the numerical spectrum, we prove a discrete compactness property for the corresponding DG space. The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations. We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals.     

Failure resistant topology optimization of structures using nonlocal elastoplastic-damage model

For energy absorbing structures made up of ductile materials, the plastic strain accumulation often leads to early material damage and failure, which can deteriorate the overall structural performance. The goal of this work is to limit this damage in elastoplastic designs using the density-based topology optimization framework such that the optimized structures can absorb energy in a more controllable manner. To this end, an implicit nonlocal coupled elastoplastic damage model is considered for simulating the material damage and softening behavior. The nonlocal effect from the void elements is removed by introducing a scaling scheme for the nonlocal parameters. Path-dependent sensitivity is derived analytically using an adjoint method whose accuracy is further verified by the central difference method. The effectiveness of the proposed method is demonstrated through several numerical examples. It is shown that the load-carrying capacity, ductility, as well as ultimate plastic work dissipation capacity of the optimized design, can be considerably improved by the proposed method.     

Solution of the nonlinear elasticity imaging inverse problem: The incompressible case

We have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material. The data for this problem are the quasi-static deformation fields within the solid measured at two distinct overall strain levels. The main ingredients of our algorithm are a gradient based quasi-Newton minimization strategy, the use of adjoint equations and a novel strategy for continuation in the material parameters. In this paper we present several extensions to this algorithm. First, we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation. We achieve this by solving the forward problem using a residual-based, stabilized, mixed finite element formulation which circumvents the Ladyzenskaya-Babuska-Brezzi condition. Second, we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters, or by splitting the problem into two distinct steps. Finally, we present a new strain energy density function with an exponential stress-strain behavior that yields a deviatoric stress tensor, thereby simplifying the interpretation of pressure when compared with other exponential functions. We test the overall approach by solving for the spatial distribution of material parameters from noisy, synthetic deformation fields.     

Hybrid-displacement (Trefftz) formulation for softening materials

This paper reports on the nonlinear static analysis of 2D concrete structures using a non-conventional finite element formulation. The nonlinear behavior of the material is modelled with a continuum non-local and isotropic damage model. While the material’s behavior is linear elastic, a pure hybrid-displacement Trefftz formulation is adopted. From the point where the concrete assumes a nonlinear behavior, this approach degenerates into a hybrid-displacement formulation. The computational performance is tested by means of two numerical examples which show that the proposed model predicts correctly the global behavior of the structures.     

Multiphase Soft Segmentation with Total Variation and <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> Regularization

In this paper, we propose a variational soft segmentation framework inspired by the level set formulation of multiphase Chan-Vese model. We use soft membership functions valued in [0,1] to replace the Heaviside functions of level sets (or characteristic functions) such that we get a representation of regions by soft membership functions which automatically satisfies the sum to one constraint. We give general formulas for arbitrary N-phase segmentation, in contrast to Chan-Vese’s level set method only 2 ^m -phase are studied. To ensure smoothness on membership functions, both total variation (TV) regularization and H ¹ regularization used as two choices for the definition of regularization term. TV regularization has geometric meaning which requires that the segmentation curve length as short as possible, while H ¹ regularization has no explicit geometric meaning but is easier to implement with less parameters and has higher tolerance to noise. Fast numerical schemes are designed for both of the regularization methods. By changing the distance function, the proposed segmentation framework can be easily extended to the segmentation of other types of images. Numerical results on cartoon images, piecewise smooth images and texture images demonstrate that our methods are effective in multiphase image segmentation.     

Solving fracture problems using an asymptotic numerical method

The present work deals with the use of asymptotic numerical methods (ANM) to manage crack onset and crack growth in the framework of continuum damage mechanics (CDM). More specifically, an application of regularization techniques to a 1D cohesive model is proposed. The standard “triangle” damageable elastic model, which is often used in finite element codes to describe fracture of brittle materials, was chosen. Results associated with the load–unload cycle showed that ANM is convenient for numerically taking this specific irregular behavior into account. Moreover, the present paper also shows that the chosen damageable interface model can be introduced in the generalized standard material formalism, thus unabling us to define a complete energy balance associated with the damage process. In such a framework, the damage state is described by a new displacement variable. Finally, a 1D finite element application to a simple elastic damageable structure is shown to highlight the potential of this approach.     

Numerical calculation of the runaway electron distribution function and associated synchrotron emission

Synchrotron emission from runaway electrons may be used to diagnose plasma conditions during a tokamak disruption, but solving this inverse problem requires rapid simulation of the electron distribution function and associated synchrotron emission as a function of plasma parameters. Here we detail a framework for this forward calculation, beginning with an efficient numerical method for solving the Fokker–Planck equation in the presence of an electric field of arbitrary strength. The approach is continuum (Eulerian), and we employ a relativistic collision operator, valid for arbitrary energies. Both primary and secondary runaway electron generation are included. For cases in which primary generation dominates, a time-independent formulation of the problem is described, requiring only the solution of a single sparse linear system. In the limit of dominant secondary generation, we present the first numerical verification of an analytic model for the distribution function. The numerical electron distribution function in the presence of both primary and secondary generation is then used for calculating the synchrotron emission spectrum of the runaways. It is found that the average synchrotron spectra emitted from realistic distribution functions are not well approximated by the emission of a single electron at the maximum energy.     

Model reduction and control of reactor–heat exchanger networks

This paper focuses on the dynamics and control of process networks consisting of a reactor connected with an external heat exchanger through a large material recycle stream that acts as an energy carrier. Using singular perturbation arguments, we show that such networks exhibit a dynamic behavior featuring two time scales: a fast one, in which the energy balance variables evolve, and a slow time scale that captures the evolution of the terms in the material balance equations. We present a procedure for deriving reduced-order, non-stiff models for the fast and slow dynamics, and a framework for rational control system design that accounts for the time scale separation exhibited by the system dynamics. The theoretical developments are illustrated with an example and numerical simulation results.     

Efficient numerical integration of an elastic–plastic damage law within a mixed velocity–pressure formulation

This study focuses on numerical integration of constitutive laws in numerical modeling of cold materials processing that involves large plastic strain together with ductile damage. A mixed velocity–pressure formulation is used to handle the incompressibility of plastic deformation. A Lemaitre damage model where dissipative phenomena are coupled is considered. Numerical aspects of the constitutive equations are addressed in detail. Three integration algorithms with different levels of coupling of damage with elastic–plastic behavior are presented and discussed in terms of accuracy and computational cost. The implicit gradient formulation with a non-local damage variable is used to regularize the localization phenomenon and thus to ensure the objectivity of numerical results for damage prediction problems. A tensile test on a plane plate specimen, where damage and plastic strain tend to localize in well-known shear bands, successfully shows both the objectivity and effectiveness of the developed approach.     

A robust computational approach for dry powders under quasi-static and transient impact loadings

Powders are challenging materials for many engineers and scientists, since they often show unexpected behavior which is quite different from the behavior of gases, liquids or solids. In addition, powder-like materials appear quite often in real applications. Our study is driven by the need to appropriately grasp the behavior of dry powder and our main focus is on the damping and energy absorbing behavior of dry powder under impact loading. The overall approach, including both the model and the algorithmic setup, that we developed for this purpose is presented in this paper. Applicability to quasi-static as well as highly transient real world problems and robustness are crucial constraints for the whole undertaking. These requirements are met through a model with relatively few and, more importantly, easy-to-obtain material parameters and through some special algorithmic developments.After a general introduction into powder, our continuum model based on finite strain elasto-plasticity for the simulation of quasi-static and transient dynamic processes is presented. Then the algorithmic setup, i.e. the required return mapping algorithm formulated in principal stresses, is presented. Finally, the parameter determination from standard laboratory tests is described and appropriate numerical results are shown for both quasi-static and highly transient impact cases.     

Fracture Patterns Design for Anisotropic Models with the Material Point Method

Physically plausible fracture animation is a challenging topic in computer graphics. Most of the existing approaches focus on the fracture of isotropic materials. We proposed a frame-field method for the design of anisotropic brittle fracture patterns. In this case, the material anisotropy is determined by two parts: anisotropic elastic deformation and anisotropic damage mechanics. For the elastic deformation, we reformulate the constitutive model of hyperelastic materials to achieve anisotropy by adding additional energy density functions in particular directions. For the damage evolution, we propose an improved phase-field fracture method to simulate the anisotropy by designing a deformation-aware second-order structural tensor. These two parts can present elastic anisotropy and fractured anisotropy independently, or they can be well coupled together to exhibit rich crack effects. To ensure the flexibility of simulation, we further introduce a frame-field concept to assist in setting local anisotropy, similar to the fiber orientation of textiles. For the discretization of the deformable object, we adopt a novel Material Point Method(MPM) according to its fracture-friendly nature. We also give some design criteria for anisotropic models through comparative analysis. Experiments show that our anisotropic method is able to be well integrated with the MPM scheme for simulating the dynamic fracture behavior of anisotropic materials.     

Numerical instabilities in level set topology optimization with the extended finite element method

This paper studies level set topology optimization of structures predicting the structural response by the eXtended Finite Element Method (XFEM). In contrast to Ersatz material approaches, the XFEM represents the geometry in the mechanical model by crisp boundaries. The traditional XFEM approach augments the approximation of the state variable fields with a fixed set of enrichment functions. For complex material layouts with small geometric features, this strategy may result in interpolation errors and non-physical coupling between disconnected material domains. These defects can lead to numerical instabilities in the optimized material layout, similar to checker-board patterns found in density methods. In this paper, a generalized Heaviside enrichment strategy is presented that adapts the set of enrichment functions to the material layout and consistently interpolates the state variable fields, bypassing the limitations of the traditional approach. This XFEM formulation is embedded into a level set topology optimization framework and studied with “material-void” and “material-material” design problems, optimizing the compliance via a mathematical programming method. The numerical results suggest that the generalized formulation of the XFEM resolves numerical instabilities, but regularization techniques are still required to control the optimized geometry. It is observed that constraining the perimeter effectively eliminates the emergence of small geometric features. In contrast, smoothing the level set field does not provide a reliable geometry control but mainly improves the convergence rate of the optimization process.     

A Bernstein-Bézier Based Approach to Soft Tissue Simulation

This paper discusses a Finite Element approach for volumetric soft tissue modeling in the context of facial surgery simulation. We elaborate on the underlying physics and address some computational aspects of the finite element discretization.
In contrast to existing approaches speed is not our first concern, but we strive for the highest possible accuracy of simulation. We therefore propose an extension of linear elasticity towards incompressibility and nonlinear material behavior, in order to describe the complex properties of human soft tissue more accurately. Furthermore, we incorporate higher order interpolation functions using a Bernstein-Bézier formulation, which has various advantageous properties, such as its integral polynomial form of arbitrary degree, efficient subdivision schemes, and suitability for geometric modeling and rendering. In addition, the use of tetrahedral Finite Elements does not put any restriction on the geometry of the simulated volumes.
Experimental results obtained from a synthetic block of soft tissue and from the Visible Human Data Set illustrate the performance of the envisioned model.     

Remarks on the asymptotic behaviour of Koiter shells

We review some results about the behaviour of a general Koiter shell, in the framework of linear elasticity. In particular, we investigate the asymptotics (for the thickness tending to zero) of the energy functional and of the percentage of the energy which is stored in the bending term. Such an analysis is motivated by the need to better understand how to numerically treat an arbitrary thin shell, when the discretization is performed using a finite element strategy. We present some instances to which our theory can be applied. Some numerical tests confirming our theoretical predictions are also provided.     

Multiphase material optimization for fiber reinforced composites with strain softening

The present paper addresses an optimization strategy of textile fiber reinforced concrete (FRC) with emphasis on its special failure behavior. Since both concrete and fiber are brittle materials a prominent objective for FRC structures is concerned with the improvement of ductility. Despite above unfavorable characteristics the interface between fiber and matrix plays a substantial role in the structural response. This favorable ‘composite effect’ is related to material parameters involved in the interface and the material layout on the small scale level. Therefore the purpose of the present paper is to improve the structural ductility of FRC at the macroscopic level applying an optimization method with respect to significant material parameters at the small scale level. The method discussed is based on multiphase material optimization. This methodology is extended to a damage formulation. The performance of the proposed method is demonstrated in a series of numerical examples; it is verified that the ductility can considerably be improved.