2D Examples Research Articles

GeosPIn: A MATLAB library for geostatistical petrophysical inversion

The petrophysical inversion of seismic data is one of the key components of seismic reservoir characterization. The goal of petrophysical inversion is to estimate the petrophysical properties of reservoir rocks, such as porosity, volumes of minerals or lithologies, and water and hydrocarbon saturations, from seismic data. This process can be performed by combining seismic amplitude-variation-with-offset (AVO) modeling and rock-physics relations with deterministic or probabilistic inverse theory methods, either in a two-step approach based on seismic AVO inversion and rock-physics inverse mapping or in a single-loop step that includes both. We present a comprehensive MATLAB library named Geostatistical Petrophysical Inversion. The novelty of the library lies in a single-loop geostatistical inversion method based on probability field simulation and ensemble smoother multiple data assimilation, as well as its implementation for applications to 3D data sets. The library also includes a Bayesian petrophysical inversion based on the two-step approach for comparison. We illustrate the main algorithms, explain the structure of the source codes, and demonstrate the library applications through 1D and 3D examples.

A proportional topology optimization method with level-set description and evolutionary strategy

In this work, we propose a proportional topology optimization method with a level-set description and evolutionary strategy (PTO-LSES). A level-set function (LSF) evolution scheme based on the nodal compliance proportion is presented and utilized in the new method. Furthermore, a compliance proportion filtering technique, a regularization scheme, and a material interpolation model with penalty are also used in the proposed method. To determine the effectiveness and superiority of the PTO-LSES method, both the 2D (cantilever beam) and 3D (cantilever and half-MBB beams) numerical examples solving the compliance minimization problem, in contrast to the conventional solid isotropic material with penalization (SIMP) and the original proportional topology optimization (PTO) methods, were tested. In addition, we discuss the impact of compliance proportion filtering and regularization schemes on the new method using 2D half-MBB and cantilever beams. The results showed that compared to the SIMP and PTO methods, the PTO-LSES method offered great advantages via the ability to acquire better objective function (compliance) values and more ideal topology structures with smooth and clear boundaries when subjected to specific conditions. Furthermore, applying the compliance pro-portion filtering technique or regularization scheme to the PTO-LSES method enhanced the performance of the new method in many ways.

Nonlinear fatigue damage constrained topology optimization

In engineering applications, plenty of components are subjected to variable-amplitude cyclic loading, resulting in fatigue damage, which is one of the main forms of structural damage. While the linear damage rule has long served as a fundamental approach, its limitations necessitate advancements for more accurate fatigue life predictions. Hence, this paper introduces a pioneering method termed nonlinear fatigue damage constrained topology optimization (NFDCTO). This method integrates several key components: the rainflow counting method to evaluate the non-proportional cyclic load levels, Basquin's equation to describe the S-N curve, and Morrow's plastic work interaction rule to calculate the nonlinear cumulative damage of the structure. Consequently, we establish a mathematical model for the NFDCTO method based on these components. The incorporation of penalized damage aggregation with the P-Normal function is used to address significant constraints and singularity challenges. Furthermore, by employing the adjoint method, we derive sensitivity equations for both the objective function and fatigue constraint function concerning the design variables. Subsequently, the superiority of the NFDCTO method over traditional linear fatigue damage method were verified applying 2D l-shaped beam and simply supported structures as examples. Concurrently, bridge structures were employed to investigate the effect of the sensitivity index of the material to the history of variable amplitude stresses on the optimization results. In addition, the influence of the fatigue penalty factor on the topology optimization results was assessed using a 2D cantilever beam. Finaly, we verify the effectiveness and feasibility of the NFDCTO method in addressing fatigue optimization challenges for 3D structures utilizing two 3D examples.

Simple MATLAB and Python scripts for multi-exponential analysis.

Multi-exponential decay is prevalent in magnetic resonance spectroscopy, relaxation, and imaging. This paper describes simple MATLAB and Python functions and scripts for regularized multi-exponential analysis methods for 1D and 2D data and example test problems and experiments. Regularized least-squares solutions provide production-quality outputs with robust stopping rules in ~5 and ~20 lines of code for 1D and 2D inversions, respectively. The software provides an open-architecture simple solution for transforming exponential decay data to the distribution of their decay lifetimes. Examples from magnetic resonance relaxation of a complex fluid, a Danish North Sea crude oil, and fluid mixtures in porous materials-brine/crude oil mixture in North Sea reservoir chalk-are presented. Developed codes may be incorporated in other software or directly used by other researchers, in magnetic resonance relaxation, diffusion, and imaging or other physical phenomena that require multi-exponential analysis.

Extended multiscale FEM-based concurrent optimization of three-dimensional graded lattice structures with multiple microstructure configurations

In this work, a novel concurrent design method for three-dimensional graded lattice structures considering multiple microstructure configurations is proposed. Firstly, the Multiple Variable Cutting (M-VCUT) level set method is employed to represent the geometry of lattice microstructures and define the optimization parameters. Then, the extended multiscale finite element method (EMsFEM) is implemented for calculating the equivalent stiffness matrices of microstructures in offline form. Compared with the traditional homogenization-based multiscale approaches, the proposed method eliminates the scale separation assumptions and takes into account the coupling effects in different directions of materials. In addition, a data-driven method and a parallel computing approach are integrated together so that computational efficiency can be greatly improved. After the iteration process, the optimized full-scale graded structures are reconstructed using interpolation technology to generate well-connected microstructures in neighboring cells. Finally, several 3D numerical examples are utilized to validate the effectiveness of the proposed method, where multiple lattice microstructure configurations are considered, including truss-lattices and plate-lattices.

Multi-scale concurrent topology optimization of cellular structures with multiple microstructures for minimizing dynamic response in the time domain

This paper proposes a concurrent topology optimization method for optimizing the structures that are periodically filled with multiple microstructures excited by dynamic response in the time domain. At the macroscale, different microstructures are considered as different materials. To generate the various distribution of different microstructures, a multi-material interpolation approach based on the Solid Isotropic Material with Penalization (SIMP) is integrated. At the microscale, the energy-based homogenization method (EBHM) is employed to determine the macroscopic effective properties of the microstructures. All macroscale elements with identical material are represented by a distinct microstructure. For the proportional damping model, the HHT-α is implemented as a time integration technique to obtain the dynamic response of multi-scale and assumed multi-material structures. The objective function of the topology optimization problem is to minimize the mean dynamic compliance. Combined differentiate-then-discretize method with the adjoint variable method, the sensitivity analysis applies the gradient-based Zhang-Paulino-Ramos Jr. (ZPR) algorithm or Method of Moving Asymptotes (MMA) to update the design variables under multiple constraints in the time and space-discretized system. The proposed approach is numerically performed through 2D and 3D examples to demonstrate its effectiveness.

Multiscale topology optimization of structures by using isogeometrical level set approach

This study aims to optimize topology of structures at macro and micro scales, simultaneously, by using a level set method in an isogeometric analysis (IGA) framework. To achieve this, equilibrium and homogenization equations in the model are solved by IGA method. The level set functions are defined over a grid in parameter space of associating b-splines of the IGA model. Therefore, control net of the model and level set grid are separated and there is no need to refine the control net for having smooth boundaries. Sensitivity analyses for both scales are performed to calculate the velocity of boundary points and the level set functions are updated by solving reaction-diffusion equations. Finally, several 2D and 3D examples with different geometry and boundary conditions are provided to show performance and efficiency of the method. Obtained results show good agreement with examples in literature in terms of both topology and final value of objective function. Also, by using IGA level set method, smooth boundaries are achieved in the final topology of micro and macro structures.

Engineering 3D Floquet Codes by Rewinding

Floquet codes are a novel class of quantum error-correcting codes with dynamically generated logical qubits arising from a periodic schedule of noncommuting measurements. We utilize the interpretation of measurements in terms of condensation of topological excitations and the rewinding of measurement sequences to engineer new examples of Floquet codes. In particular, rewinding is advantageous for obtaining a desired set of instantaneous stabilizer groups on both toric and planar layouts. Our first example is a Floquet code with instantaneous stabilizer codes that have the same topological order as the three-dimensional (3D) toric code(s). This Floquet code also exhibits a splitting of the topological order of the 3D toric code under the associated sequence of measurements, i.e., an instantaneous stabilizer group of a single copy of the 3D toric code in one round transforms into an instantaneous stabilizer group of two copies of the 3D toric code up to nonlocal stabilizers in the following round. We further construct boundaries for this 3D code and argue that stacking it with two copies of the 3D subsystem toric code allows for a transversal implementation of the logical non-Clifford controlled-controlled-Z gate. We also show that the coupled-layer construction of the X-cube Floquet code can be modified by a rewinding schedule such that each of the instantaneous stabilizer codes is finite depth equivalent to the X-cube model up to toric codes; the X-cube Floquet code exhibits a splitting of the X-cube model into a copy of the X-cube model and toric codes under the measurement sequence. Our final 3D example is a generalization of the 2D Floquet toric code on the honeycomb lattice to three dimensions, which has instantaneous stabilizer codes with the same topological order as the 3D fermionic toric code. Published by the American Physical Society 2024

A Methodology for Structural Damage Detection Adding Masses

ABSTRACT In engineering applications, it is standard to consider modal parameters (e.g. natural frequencies and mode shapes), to obtain measurable changes induced by damage events. However, such changes are often too subtle to detect and interpret accurately. In order to overcome this limitation and enhance the efficacy of damage detection, this study introduces an innovative technique inspired by Prof. Lekszycki, outlined prior to his untimely departure. This technique is designed to highlight the influence of structural damage on natural frequencies. In other words, we add an external mass in a general point of a structure and calculate the set of natural frequencies before and after the occurrence of a damage event. We, therefore, repeat this procedure for all the points of the structure. The hypothesis we have verified in this paper is that if the mass is posed in the neighborhood of the place where the damage is considered, then it interacts with the undamaged and with the damaged structure in a different way. In particular, the changes of the natural frequencies between the undamaged and the damaged structure are emphasized. We verify the correctness of this idea for selected and simple examples in both 2D and 3D environments.

How Impedance Measurements and Imaging Can Be Used to Characterize the Conductivity of Tissues During the Workflow of an Electroporation-Based Therapy.

In this article we investigate the possibility of using needles, which the interventional radiologist inserts near a deep-seated tumor during an electroporation-based therapy, to characterize the electrical conductivity of patient's tissues. Specifically, we propose to exploit voltage/current measurements and imaging that are performed prior to the application of electroporation pulses. The approach is partly based on the concepts of electrical impedance tomography; however, imaging is used to build a specific geometric model and compensate for the lack of information resulting from the small number of electrodes available. 3D canonical and clinical examples, where a few electrodes surround a tumor, demonstrate the feasibility of this method: solving the inverse problem to estimate tissues conductivity converges in a few iterations. For a given error on the measurement, it is also possible to calculate the error on the estimated conductivities. The uncertainty error with clinical data is at best 5% for one of the tissues identified, due to the limitations of the clinical device used. Various improvements to clinical devices are discussed to make the conductivity estimation more accurate but also to extract more information.

Algorithm 1041: HiPPIS—A High-order Positivity-preserving Mapping Software for Structured Meshes

Polynomial interpolation is an important component of many computational problems. In several of these computational problems, failure to preserve positivity when using polynomials to approximate or map data values between meshes can lead to negative unphysical quantities. Currently, most polynomial-based methods for enforcing positivity are based on splines and polynomial rescaling. The spline-based approaches build interpolants that are positive over the intervals in which they are defined and may require solving a minimization problem and/or system of equations. The linear polynomial rescaling methods allow for high-degree polynomials but enforce positivity only at limited locations (e.g., quadrature nodes). This work introduces open-source software (HiPPIS) for high-order data-bounded interpolation (DBI) and positivity-preserving interpolation (PPI) that addresses the limitations of both the spline and polynomial rescaling methods. HiPPIS is suitable for approximating and mapping physical quantities such as mass, density, and concentration between meshes while preserving positivity. This work provides Fortran and Matlab implementations of the DBI and PPI methods, presents an analysis of the mapping error in the context of PDEs, and uses several 1D and 2D numerical examples to demonstrate the benefits and limitations of HiPPIS.

A Wavelet Extraction Method of Attenuation Media for Direct Acoustic Impedance Inversion in Depth Domain

The seismic image produced by pre-stack depth migration is more accurate and has clearer geological significance than the time image. However, the waveform of the depth-domain seismic image is affected not only by depth-dependent velocity variation but also by media attenuation, resulting in strong spectral variation of depth-domain seismic data. Therefore, depth-domain seismic inversion is still challenging. We propose a wavelet extraction method of attenuation media based on the generalized seismic wavelet, to address this issue. Then, the estimated depth-domain wavelets were applied to the direct acoustic impedance inversion. First, we investigated the effect of attenuation media on depth-domain source wavelets and derived an analytical formula for the depth-domain wavelets of attenuation media. Next, the time-domain generalized seismic wavelet was extended to the depth domain, which was utilized to study the feasibility of using the generalized seismic wavelet to characterize the seismic wavelet of the depth-domain attenuation media. Based on the orthogonal matching pursuit, we propose a method to extract the depth-domain generalized seismic wavelet directly from depth-domain seismic data. Finally, we applied this method to the depth-domain direct acoustic impedance inversion of a 3D field data example. Tests on the synthetic and 3D field datasets show that the proposed method can correctly extract the depth-domain seismic wavelet of attenuation media and attain direct inversion of the depth-domain acoustic impedance with high accuracy. Therefore, our method is effective and has robust potential in reservoir characterization, fluid prediction, and attribute extraction in the depth domain.

Robust contact-constrained topology optimization considering uncertainty at the contact support

In this paper, the general framework for contact-constrained topology optimization of Strömberg and Klarbring (2010) is extended to robust topology optimization. In doing so, a linear elastic design domain is considered and the augmented Lagrangian approach is used to model the unilateral contact. For topology optimization, the design space is parametrized with the SIMP-approach and the Sigmund’s filter is applied. Additionally, the robust framework considers uncertainties at the contact support such as deviations of the geometry of the contact surface and the friction coefficient. Both uncertainties are described by the first-order second-moment method which leads to minimal additional costs. In fact, only two additional linear equations must be solved to obtain the robust objective and its gradient with respect to the design variables. Having both the objective and the gradient, the design update is computed with the method of moving asymptotes. The robust framework is applied to 2D and 3D examples to prove its scalability for real-world applications.

Elastoplastic peridynamic formulation for materials with isotropic and kinematic hardening

We present an ordinary state-based peridynamic model in 2D and 3D consistent with rate-independent J2 plasticity with associated flow rule. The new contribution is the capability of the elastoplastic law to describe isotropic, kinematic and mixed hardening. The hardening formulations follow those available in the literature for classical elastoplasticity. The comparison between the results obtained with the peridynamic model and those obtained with a commercial FEM software shows that the two approaches are in good agreement. The extent of the plastic regions and von Mises stress computed with the new model for 2D and 3D examples match well those obtained with FEM-based solutions using ANSYS.

A Bloch analysis extended to weakly disordered periodic media

This paper develops an asymptotic method to predict the dispersion curves of a weakly disordered medium. The disordered coefficients of the original acoustic equation are related to periodic coefficients through a change of variables, assumed asymptotically close to the identity. At leading order in the amplitude of the disorder, the medium is driven by an acoustic equation with periodic coefficients, which can be analyzed through a classical Floquet–Bloch approach. At first order, with simple eigenvalues, a simple post-processing of the periodic eigenvalues and eigenfunctions of the leading order allows to account for the weakly disordered character of the original coefficients. In the case of repeated eigenvalues (for instance, at Dirac points), a residual is introduced, whose minimization allows to recover the opening of the band gaps with the loss of periodicity. Examples in 1D and 2D illustrate the validity of our asymptotic approach, by comparing its results with eigenmodes and dispersion curves computed over much larger periods for reference.

A mixed Fourier-variational approach to solve differential or integro-differential wave equations for magnetised plasmas

The All ORders Spectral Algorithm (AORSA) wave equation solver by Jaeger (Jaeger et al 2001 Phys. Plasmas 8 1573) solves the integro-differential wave equation relevant for the radio frequency (RF) domain and for fusion-relevant conditions in tokamaks or stellarators, retaining all finite Larmor radius corrections by substituting the continuous Fourier integrals by a sum over a discrete set of modes. Its strength is also its weakness: the simplicity of the method results in significant computational effort, a full matrix needing to be inverted to solve the associated linear system. Based on the notion that modes are gradually more independent if their eigenvalues differ, the present paper proposes a straightforward numerical method to partly alleviate this need, allowing to substitute the full system matrix by a banded one. The adopted method can be applied to a wide variety of equations. A few 1D examples—of relevance for solving the wave equation in the RF domain of frequencies—are provided: the tunneling equation is used to illustrate the potential of the method, and the all-FLR wave equation (retaining all Finite Larmor Radius corrections in the dielectric response) adopted by Jaeger is solved comparing the solutions found to those based on simpler models (a cold plasma and a ‘tepid plasma’ - i.e. a kinetic model truncated at zero order in Larmor radius—description).

Topology optimisation for vat photopolymerization 3D printing of ceramics with flushing jet accessibility constraint

ABSTRACT For vat photopolymerization 3D printing technology with high-viscosity ceramic slurry, the cleaning of residual uncured slurry presents a challenging task for lattice-based porous structures. Hence, this study contributes a multi-scale topology optimisation method for lattice-solid hybrid structures dedicated to ceramic vat photopolymerization 3D printing. A novel flushing jet accessibility constraint is proposed, aiming to effectively remove the residual uncured slurry after finishing the vat photopolymerization. Specifically, a multi-material interpolation for graded lattices and solid materials is introduced to form the fundamental of topology optimisation. Then, a series of filtering and projection operations are developed to ensure the accessibility of flushing waters to every location that may have residual ceramic slurry. Accordingly, a multi-scale topology optimisation mathematical model is established and solved with the adjoint sensitivities. 2D and 3D numerical examples are conducted to evaluate the effectiveness of the proposed method. Furthermore, a 3D printing experiment is performed to validate the full water jet accessibility of the optimised structural solution and one pillow bracket part is carried out as a practical application demonstration.

Concurrent multiscale and multi‐material optimization method for natural vibration design of porous structures

AbstractThis paper proposes a concurrent multiscale optimization method of macrostructure topology and microstructure shapes for porous structures, aimed at maximizing a specified natural frequency. The multi‐material distribution of the macrostructure and the shape of the microstructures are optimized by topology and shape optimization, respectively. The homogenized properties of the porous materials are calculated using the homogenization method, and the homogenized elastic tensor and density are applied to the macrostructure. The optimum distribution of the porous material in the macrostructure is determined by a multi‐material topology optimization using the generalized solid isotropic material with penalization (GSIMP) method, which is a method to obtain multi‐material topology by implementing the successive binarization. The KS (Kreisselmeier–Steinhauser) function is introduced to solve the repeated natural frequencies issue that lies in the maximization of a specified natural frequency. An area‐constrained multiscale optimization problem is formulated as a distributed‐parameter optimization problem, and the sensitivity functions are derived using the Lagrange multiplier method and the adjoint variable method. Based on the obtained sensitivity functions, the design variables are updated using the H1 gradient method (i.e., a nonparametric shape/topology optimization method). The effectiveness of the proposed method for maximizing a specified natural frequency is confirmed through 2D and 3D numerical examples, in which the influences of sub‐regions of the macrostructure and anisotropic material of the microstructures are also investigated and the results are discussed.

Some Error Bounds for 2‐Point Right Radau Formula in the Setting of Fractional Calculus via s‐Convexity

In this paper, we present a new approach to construct fractional 2‐point right Radau type integral inequalities using a novel identity, for functions with s‐convex first derivatives in the second sense via Riemann–Liouville fractional integral operators. We then demonstrate the accuracy of our results through a 2D example, as well as practical applications of the integral inequalities to quadrature formulas and special means such as arithmetic and p‐logarithmic means.

A novel delayed exponent coupled chaotic map with countering dynamical degradation

<abstract> <p>While chaotic systems have found extensive applications across diverse scientific domains due to their inherent advantages, they often degrade into cyclic patterns when simulated on hardware with limited computational precision. This results in a pronounced decline in properties related to chaotic dynamics. To address this issue, we introduce the delayed exponent coupled chaotic map (DECCM). This model is designed to enhance the chaotic dynamics of the original map, especially at lower computational precisions. Additionally, DECCM can transform any proficient 1-dimensional seed map into an <italic>N</italic>-dimensional chaotic map. Extensive simulation and performance tests attest to the robust chaotic characteristics of our approach. Furthermore, DECCM holds distinct advantages over premier algorithms, particularly in period analysis experiments. We also introduce various seed maps into DECCM to present 2D and 3D examples, ensuring their generalization through relevant performance evaluations.</p> </abstract>