Numerical Tools Research Articles

Study of Hysteresis Losses in Partial Penetration Oscillations Through Experiment, Analytical Algorithm and New Numerical Tool

Study of Hysteresis Losses in Partial Penetration Oscillations Through Experiment, Analytical Algorithm and New Numerical Tool

Numerical modeling of wave interaction with a porous floating structure consisting of uniform spheres

Porous materials are increasingly being used in the design of floating structures in coastal and ocean engineering, but there is a lack of numerical tools that can aid in the design of a movable floating porous structure. To close this gap, the existing volume-averaged numerical model for flow interacting with a fixed porous body was extended to floating scenarios by (1) using the relative velocity in the porous friction force, (2) calculating rigid body motion using volume integral of porous body force, and (3) modifying a dynamic mesh algorithm for a mobile porous body. As a demonstration, the developed model was applied to a porous floating structure consisting of cubically packed uniform spheres. Two sets of model applications were involved. The first set considered three-dimensional flow around a fixed porous block placed beneath the free surface. The measured total force on the block under wave or steady flows was predicted accurately with an error less than 10%. The second set involved a two-dimensional wave interacting with a floating porous block representing a breakwater. For free-floating conditions, the model can accurately predict the dynamic response of the structure, including the time varying movement of its rigid body and the mean drift. For mooring-restrained conditions, the mooring force and wave transmission coefficients were also predicted well with an error less than 20%. The proposed numerical approach can be applied to other floating structures with a rigid volumetric porous body. Future research is also required to study the microscopic pore flows, upon which more detailed parameterization of the porous media can be derived.

Efficient identification of geometric errors in CNC machine tools based on dual-frequency laser interferometry

In contemporary production, the computer numerical control machine tool is an essential processing apparatus. However, its geometric errors can often impact processing accuracy and stability. Therefore, an innovative geometric error identification method for CNC machine tools is proposed, which uses the polarization information in the dual-frequency laser interferometer to improve the measurement accuracy. By optimizing the polarization state of the laser system, the ability to identify the geometric error of the machine tool is improved. The findings of this study indicated that the measurement accuracy, measurement range, ease of operation, reliability, cost and applicability of the dual-frequency laser interferometer-based geometric error identification method for computer numerical control machine tools were 0.91, 0.87, 0.93, 0.77, 0.94, 0.85 and 0.97, respectively, which were better than the comprehensive performance of other methods. The study's suggested method offers a solid foundation for raising the precision and machining quality of machine tools by effectively and precisely identifying the geometric faults of CNC tools. The study's findings also establish the groundwork for future widespread use of dual-frequency laser interferometers in the detection of geometric faults in computer numerical control machine tools by offering theoretical justification for real-world uses.

Prediction of retrogressive landslide in sensitive clays by incorporating a novel strain softening law into the Material Point Method

Sensitive clays, when subjected to large strains, exhibit a unique strain-softening behavior, transforming into a remolded liquid with remarkably low shear strength. When a slope fails, this behavior leads to the remolded clay moving away from its original position, triggering subsequent failures and catastrophic outcomes. To accurately predict such scenarios, it is crucial to incorporate realistic strain-softening characteristics into the constitutive soil model. This paper presents a novel yet practical strain-softening law developed by the authors that effectively captures the post-peak behavior of sensitive clays down to their remolded strength. The softening law is implemented in an elastoplastic Mohr-Coulomb model and incorporated into Anura3D, an open-source software that uses the Material Point Method to simulate large deformations. The constitutive model is calibrated by simulating the stress-strain behavior through direct shear tests conducted at three sensitive clay landslide locations. The accuracy of the overall numerical framework is assessed by predicting the post-failure movements of these landslides. Notably, two of the landslides, Sainte-Monique (1994) and Saint-Jude (2010), have previously been analyzed using other numerical tools, allowing for a comparative analysis with the method presented here. The third landslide, the Saint-Luc-de-Vincennes landslide, representing a composite flow slide and spread, has been numerically simulated for the first time. The post-failure behavior observed in the landslide events is compared with field observations and other numerical analyses. The results show that the MPM models with the proposed strain-softening law can reasonably predict post-failure retrogression and runout distance, which are crucial parameters for determining the risk of landslides in sensitive clays.

High accuracy solutions for the Pochhammer–Chree equation in elastic media

This study investigates the nonlinear Pochhammer–Chree equation, a model crucial for understanding wave propagation in elastic rods, through the application of the Khater III method. The research aims to derive precise analytical solutions and validate them using He’s variational iteration method (VIM). The Pochhammer–Chree equation’s relationship to other nonlinear evolution equations, such as the Korteweg-de Vries and nonlinear Schrödinger equations, underscores its significance in the field of nonlinear wave dynamics. The methodology employs the Khater III method for deriving analytical solutions, while He’s VIM serves as a numerical validation tool, ensuring the accuracy and stability of the obtained results. This dual approach not only yields novel solutions but also provides a robust framework for analyzing complex wave phenomena in elastic media. The findings of this study have significant implications for material science and engineering applications, offering new insights into the behavior of waves in elastic rods. By bridging the gap between theoretical models and practical applications, this research contributes to the advancement of both mathematical theory and physical understanding of nonlinear wave dynamics. Situated within the domain of applied mathematics, with a focus on nonlinear wave equations, this work exemplifies the interdisciplinary nature of contemporary research in mathematical physics. The results presented herein open new avenues for future investigations in related fields and highlight the potential for innovative applications in material science and engineering.

When and why microbial-explicit soil organic carbon models can be unstable

Abstract. Microbial-explicit soil organic carbon (SOC) cycling models are increasingly being recognized for their advantages over linear models in describing SOC dynamics. These models are known to exhibit oscillations, but it is not clear when they yield stable vs. unstable equilibrium points (EPs) – i.e., EPs that exist analytically but are not stable in relation to small perturbations and cannot be reached by transient simulations. The occurrence of such unstable EPs can lead to unexpected model behavior in transient simulations or unrealistic predictions of steady-state soil organic carbon (SOC) stocks. Here, we ask when and why unstable EPs can occur in an archetypal microbial-explicit model (representing SOC, dissolved OC (DOC), microbial biomass, and extracellular enzymes) and some simplified versions of it. Further, if a model formulation allows for physically meaningful but unstable EPs, can we find constraints in the model parameters (i.e., environmental conditions and microbial traits) that ensure stability of the EPs? We use analytical, numerical, and descriptive tools to answer these questions. We found that instability can occur when the resupply of a growth substrate (DOC) is (via a positive feedback loop) dependent on its abundance. We identified a conservative, sufficient condition in terms of model parameters to ensure the stability of EPs. Principally, three distinct strategies can avoid instability: (1) neglecting explicit DOC dynamics, (2) biomass-independent uptake rate, or (3) correlation between parameter values to obey the stability criterion. While the first two approaches simplify some mechanistic processes, the third approach points to the interactive effects of environmental conditions and parameters describing microbial physiology, highlighting the relevance of basic ecological principles for the avoidance of unrealistic (i.e., unstable) simulation outcomes. These insights can help to improve the applicability of microbial-explicit models, aid our understanding of the dynamics of these models, and highlight the relation between mathematical requirements and (in silico) microbial ecology.

Two-scale concurrent simulations for crack propagation using FEM–DEM bridging coupling

The Discrete element method (DEM) is a robust numerical tool for simulating crack propagation and wear in granular materials. However, the computational cost associated with DEM hinders its applicability to large domains. To address this limitation, we employ DEM to model regions experiencing crack propagation and wear, and utilize the finite element method (FEM) to model regions experiencing small deformation, thus reducing the computational burden. The two domains are linked using a FEM–DEM coupling, which considers an overlapping region where the deformation of the two domains is reconciled. We employ a “strong coupling” formulation, in which each DEM particle in the overlapping region is constrained to an equivalent position obtained by nodal interpolation in the finite element. While the coupling method has been proved capable of handling propagation of small-amplitude waves between domains, we examine in this paper its accuracy to efficiently model for material failure events. We investigate two cases of material failure in the DEM region: the first one involves mode I crack propagation, and the second one focuses on rough surfaces’ shearing leading to debris creation. For each, we consider several DEM domain sizes, representing different distances between the coupling region and the DEM undergoing inelasticity and fracture. The accuracy of the coupling approach is evaluated by comparing it with a pure DEM simulation, and the results demonstrate its effectiveness in accurately capturing the behavior of the pure DEM, regardless of the placement of the coupling region.

Chaotic libration and control of a space tether-sail system in Earth polar orbits with J2 perturbation

The recently proposed space tether-sail system could enable various interesting and important potential space missions thanks to its thin and long connecting tether and solar radiation pressure (SRP) force without consuming propellants. This paper focuses on nonlinear libration and its control of a tether-sail system in Earth polar orbits. Firstly, nonlinear coupled in-plane and out-of-plane dynamics of the system with a rigid mass-distributed tether and two point masses (a chief satellite and sail respectively) is established using the Lagrangian equation method. Secondly, the predicted occurrence of chaotic in- and out-of-plane librational motions is verified utilizing numerical tools such as presenting the time history of libration, the phase plane, the Poincaré section and power spectral density(PSD). Specifically, the paper investigates the sensitivity of nonlinear dynamical responses to initial values, focuses on the chaotic motion caused by the coupling between the in- and out-of-plane librations, the J2 perturbation of the Earth gravitational field and the small eccentricity of the orbits, also studies the nonlinear librational characteristics influenced by the initial mechanical energy (the Hamiltonian). Thirdly, chaotic librational motions will be controlled merely using modulatable SRP force by the sail propellantlessly. A sliding mode controller based on exponential approaching law is developed. To mitigate the effects of control torque saturation, an input compensator based on Radial Basis Function (RBF) neural network is designed. In addition, to address the cumbersome and inefficient manual tuning of control parameters for the sliding mode controller, a parameter tuning algorithm based on Learning Automata is designed. This algorithm is capable of tuning a high-quality set of controller parameters within 100 iterations. The effectiveness of the propellantless actuators for chaotic libration motion control and developed control algorithm is supported by numerical simulations.

Mitigating density fluctuations in particle-based active nematic simulations

Understanding active matter has led to new perspectives on biophysics and non-equilibrium dynamics. However, the development of numerical tools for simulating active fluids capable of incorporating non-trivial boundaries or inclusions has lagged behind. Active particle-based methods, which typically excel at this, suffer from large density fluctuations that affect the dynamics of inclusions. To this end, we advance the Active-Nematic Multi-Particle Collision Dynamics algorithm, a particle-based method for simulating active nematics, by addressing the large density fluctuations that arise from activity. This paper introduces three activity formulations that mitigate the coupling between activity and local density. Local density fluctuations are decreased to a level comparable to the passive limit while retaining active nematic phenomenology and increasing the active turbulence regime four-fold in two dimensions. These developments extend the technique into a flexible tool for modeling active systems, including solutes and inclusions, with broad applications for the study of biophysical systems.

Modelling and optimization method for energy saving of computer numerical control machine tools under operating condition

The widespread application of machine tools in industry results in very high energy consumption. In order to save energy, it is an effective means to reduce the processing power while improving the processing efficiency of machine tools. Since milling parameters directly affect machine performance, the selection of suitable milling parameters is particularly important. In this regard, this study proposes an integrated modelling and optimization method for energy saving of computer numerical control (CNC) machine tools under operating condition. Firstly, the operating condition of the CNC machine are analyzed and the processing power and efficiency mathematical models are established based on the four milling parameters (n, fz, ae, ap). Subsequently, the Pareto optimal solution is introduced to establish an optimization model integrating the processing power and time. To solve the optimization problem, an adaptive chaotic multi-objective chimp algorithm with fast convergence and satisfactory optimization is proposed considering three aspects: search space, convergence factor, and chaotic mapping. Afterwards, further degrees of freedom and sensitivity analyses were carried out on the constructed model using the Sobol method. Finally, comparative analysis and validation are conducted through simulation cases and actual processing experiments, respectively. The results show that both validation scenarios have the same conclusion. Compared with three existing algorithms, the proposed method reduces the processing power by 5.63 %, 4.65 % and 4.51 % and improves the processing efficiency by 27.88 %, 15.11 % and 15.31 %, respectively. Based on the above enhancements, the energy consumption is reduced by 58.21 %, 45.54 % and 41.45 % respectively.

Analysis of creep crack growth in bonded joints based on a Paris' law-like approach

Creep crack growth is one of the factors that could affect the durability of a bonded joint and compromise the safety of structures over long periods of time. However, numerical tools and test methods relating to creep crack growth in bonded joints are not widely available yet. In this work, a Creep Crack Growth Model (CCGM) is proposed for adhesively bonded joints. The model describes the relation between the crack growth rate and the energy release rate. The CCGM is fitted with results from Roller Wedge Driven creep crack growth (RWDC) tests and validated against the results obtained from tapered double cantilever beam (TDCB) tests with a constant load applied. Additionally, it is demonstrated that the proposed model can be introduced in a fatigue tool commercially available from the finite element method (FEM) code Abaqus to predict creep crack growth. The FEM results show that the phenomenological expression fitted from experimental tests results, which is used as input for the FEM tool, is reproduced with accuracy. Moreover, it is shown that the CCGM is capable of predicting creep crack growth rates when a different specimen geometry is used, thus demonstrating that the model correctly captures the physics of the problem.

A bilateral semi-resolved CFD-DEM approach for cost-effective modelling in a rotary drum

It is of significance to numerically describe particle-flow interphase flow details at a feasible computational cost. Inspired by the semi-resolved Computational Fluid Dynamics-Discrete Element Method (CFD-DEM), an improved bilateral semi-resolved CFD-DEM approach (BSM) approach is developed in this work. Compared with the conventional divided particle volume method (DPVM) and recent semi-resolved CFD-DEM (SM) approach, the BSM approach allows a higher-resolution mesh and achieves higher accuracy as well as applicability when simulating the transient flow with sharp gradients of solid volume fraction and velocity etc. Then the BSM approach is applied to a rotary drum to demonstrate the effectiveness by investigating the internal hydrodynamics details. The typical flow behaviours are detailed; then the effects of the restitution and friction on the internal flow are analysed with details to demonstrate its effectiveness in terms of repose angle, active–passive zone, solid residence time, particle mixing and axial dispersion. The results show a positive correlation of the active depth, mixing degree, and particle dispersion with the friction, while restitution increasing depicts no significant effects on this particulate flow. This work provides an improved numerical tool for understanding the multiphase transient flows involving sharp gradients.

On the numerical simulation of a phase change material melting inside periodic structures: from validation to optimization

Nowadays, numerical models are considered powerful design tools, which are largely used in several engineering applications. When applied to heat transfer applications, they have already been shown to be reliable, especially in single phase systems. When considering two-phase heat transfer, and in particular solid-liquid phase change processes, the numerical tools must be carefully handled. Thus, this paper follows the critical literature on the application of numerical methods to simulate and optimize latent thermal energy storages to show how, in most cases, the common geometrical and numerical approximations should not be applied because they might lead to misleading solutions. Starting from reliable experimental results collected during melting process of a paraffin wax inside 3D periodic structures, different simulation approaches are proposed, and their advantages and disadvantages are discussed. The results show that, when dealing with certain complex 3D structures, 2D approximations cannot reliably capture the phase-change phenomenon. The validated tool also allowed to identify the optimum pore size to maximize the solid-liquid phase change. This work can be considered as a reference for future numerical studies on the solid-liquid process inside 3D structures.

Trajectory, fate, and magnitude of continental microplastic loads to the inner shelf: A case study of the world's largest coastal shallow lagoon

The Patos Lagoon estuary is a highly significant ecosystem where freshwater from a vast and densely populated area continuously flows into the Atlantic Ocean by coastal plumes, exporting not only freshwater but also sediment, nutrients, plastics, and other contaminants. In this work, numerical modeling tools together with field data were used to assess for the first time the capacity of the coastal plume to export microplastics (MPs) to the inner shelf under different hydrodynamic conditions. Two field surveys were conducted during plume events to quantify MP concentrations and validate the model approach. A bottom-up approach was employed to estimate the potential MP export from the estuary's domain to the Atlantic Ocean. MP concentration in surface plume waters ranged from 0.20 items m−3 to 1.37 items m−3, confirmed by FTIR as synthetic polymers in a 90 %, being Polypropylene (PP) and Polyethylene (PE) the most abundant in a 73 %. The accumulation pattern was observed on the plume's frontal system, consistent with simulation results. The estimated average MP potential export rate attained 9.0 million items day−1 during moderate plume events and 47.5 million items day−1 during high discharge plume events. Strong discharge events, coupled with intense northeast winds, facilitated rapid southwestward export of MPs. Conversely, moderate to weak discharge events retained MPs closer to the estuary's mouth, enabling either longer trajectories or earlier deposition. Significant MP accumulation hotspots were identified in the gyre between the jetties and Cassino beach, as well as in the saline front within the plume boundaries. These accumulation zones may function as reservoirs for MP particles, potentially posing threats to local ecosystems. Understanding these dynamics is crucial for ongoing monitoring efforts to assess potential harmful interactions over time.

Three-dimensional elastoplastic post-buckling analysis of functionally graded plates using a novel meshfree Tchebychev-radial point interpolation approach

This paper presents a three-dimensional (3D) analysis of the post-buckling behavior of functionally graded (FG) plates for the first time using an elastoplasticity-based meshfree formulation. A novel 3D Tchebychev-radial point interpolation approach, which combines radial basis functions with Tchebychev polynomials, is developed to solve the nonlinear post-buckling problem. The incremental plastic deformation is modeled by the Prandtl–Reuss flow rule along the isotropic hardening von Mises criterion. The governing equations are derived using the principle of virtual work by considering the 3D full Green–Lagrange strain components. The Newton–Raphson technique along with the arc-length method is used to determine post-buckling equilibrium paths of FG plates. The effective elastoplastic properties of the functionally graded material are assessed by exploiting the homogenization method, named Tamura–Tomota–Ozawa (TTO) model. It has been demonstrated that the accuracy of the solution is improved by incorporating Tchebychev polynomials into the radial point interpolation method (RPIM) shape function, and the stability and robustness of the results are independent of variations in the shape parameter. Furthermore, it is confirmed that TRPIM, with a slightly higher CPU time compared to RPIM, exhibits a higher convergence rate. The excellent agreement of the results with those existing in the literature shows that the proposed meshfree method can be used as a robust and accurate numerical tool to predict the elastoplastic post-buckling behavior of FG plates. Further numerical assessments indicate that post-buckling paths are significantly affected by factors such as geometric parameters, material gradient, loading ratio, and boundary conditions (BCs).

Construction and validation of computational techniques as part of a holistic approach to complex engineering design

As a full-service provider of bespoke solutions to environmental noise problems, there is a need to have a precise understanding of the performance of our engineered products and complex assemblies. More specifically, there are risks, as well as limitations, to relying solely on a single scientific method-e.g., empirical, computational, numerical or analytical. In the following paper, we describe the validation process of sophisticated computational software-a suite of Siemens software-to bolster our risk assessment practices. Although computational tools generally require extensive 'calibration,' our interest in pursuing the software is to model diverse constraints and environmental conditions of projects, which cannot be 'simply' assessed using empirical, numerical and analytical tools. Herein are presented examples of projects utilizing our holistic approach to assuring the performance of our engineering solutions. More specifically, we employ aero-vibro-acoustical simulations to model the aero-vibro-acoustical performance (e.g., airflow and pressure drop, dynamic insertion loss) of our industrial acoustic silencers (and similar complex assemblies).

Organic solar cell efficiency improvement through architecture engineering by integrating carbon nanoring – SCAPS 1D modelling

This research work aims to investigate on improving the performance of organic solar cells (OSC) by proposing new architectures that progressively integrate carbon nanotubes. This contribution takes into account the exceptional properties of carbon nanotubes (CNT) highlighted in recent work, with the aim of addressing the current challenges of materials used in the manufacture of conventional OSCs, such as reactivity with the ambient medium, stability over time, better transport of charge carriers, etc. Using a reference OSC cell with the following structure: PEDOT/BHJ/PDINO and the numerical one-dimensional simulation tool SCAPS, which helps in validating the modelling approach, we gradually replace each layer with a CNT-based material exhibiting superior properties. In this respect, we carry out calculations on the different cells architectures, to better visualize energy-level alignment profiles, the different current-voltage (J-V), and quantum efficiency (QE) characteristics. Approaches toward efficiency improvement were discussed, by investigating the effects of active layer thickness, the doping level, the charge carriers mobility, defect density, and external parameters such as temperature. After numerous calculations and modelling, very promising results were finally achieved with a full CNT-based solar structure with the following recorded performances: VOC=0.87V, JSC=27.81mA/cm2, FF=80.91%, and PCE=19.63%. These results are the first theoretical report for a complete CNT-based modelled OSC and, given the reliability and robustness of the approach adopted, they can be considered as relevant for future theoretical and experimental work. The integration of CNT as functional layers is a remarkable step forward in OSC architectures engineering for efficiency enhancement. CNT are good candidates for light harvesting applications.

Compact Modified Elliptical-Shaped Ultra-wideband Filtering Antenna with Improved Band-Edge Selectivity

A compact filtering antenna with higher selectivity near band-edge frequencies for ultra-wideband (UWB) applications is presented and developed in this article. The antenna with filtering characteristics is formulated by integrating a rectangular-impedance resonator-based bandpass filter into the feedline of a modified elliptical-shaped monopole radiator. Initially, a modified elliptical-shaped monopole radiator is designed and numerically simulated to ensure the overall performance over the UWB band. The modified elliptical shape is used as a radiator to increase the electrical length while maintaining minimum physical size. The size of the UWB antenna is merely 0.518λ AL × 0.272 λ AL (where λAL is the guided wavelength at 2.62 GHz). Furthermore, to enhance the performance characteristics of the filtering antenna along with band-edge selectivity, geometrical parameters are optimized using a numerical simulation tool. It exhibits a simulated impedance bandwidth of 112% (3.05–10.81 GHz) with an electrical size of 0.676λ FL × 0.316λ FL (where λFL is the guided wavelength at 3.05 GHz). The experimental results of the fabricated UWB antenna and filtering antenna prototypes agree well with their simulated results.

Cantilever modelling in the railway catenary shape-finding problem

Catenary designers need accurate numerical simulation tools to improve the system performance, increase operational speed, minimize wear and prevent arcing. To obtain reliable results, the catenary numerical model must avoid over-simplifying the hypotheses, which significantly affect the outcome, even if this involves increasing the model’s complexity and computational cost.While the cantilevers that support the catenary wires are frequently omitted or over-simplified in many published studies, we incorporate the cantilevers into the catenary model to examine their impact on the static and dynamic behaviour of the catenary, and to develop a shape-finding procedure to include the cantilevers by a substructuring approach.For this we compared three scenarios: (i) a conventional catenary model without cantilevers, plus two other more realistic models that include the cantilevers: (ii) considering cantilever deformation in the geometric design, and (iii) disregarding cantilever deformation in the geometric design.The findings suggest that including the cantilevers reduces the stiffness of the system and produces a similar contact force. Although the two cantilever design options analysed result in slightly different initial catenary configurations, they also yield a very similar contact force. These findings therefore validate using simplified supports, provided they are appropriately calibrated, for which the proposed comprehensive cantilever model can be applied.

Effects of nonlinearities and geometric imperfections on multistability and deformation localization in wrinkling films on planar substrates

Compressed elastic films on soft substrates release part of their strain energy by wrinkling, which represents a loss of symmetry, characterized by a pitchfork bifurcation. Its development is well understood at the onset of supercritical bifurcation, but not beyond, or in the case of subcritical bifurcation. This is mainly due to nonlinearities and the extreme imperfection sensitivity. In both types of bifurcations, the energy–displacement diagrams that can characterize an energy landscape are non-convex, which is notoriously difficult to determine numerically or experimentally, let alone analytically. To gain an elementary understanding of such potential energy landscapes, we take a thin beam theory suitable for analyzing large displacements under small strains and significantly reduce its complexity by reformulating it in terms of the tangent rotation angle. This enables a comprehensive analytical and numerical analysis of wrinkling elastic films on planar substrates, which are effective stiffening and/or softening due to either geometric or material nonlinearities. We also validate our findings experimentally. We explicitly show how effective stiffening nonlinear behavior (e.g., due to substrate or membrane deformations) leads to a supercritical post-bifurcation response, makes the energy landscape non-convex through energy barriers causing multistability, which is extremely problematic for numerical computation. Moreover, this type of nonlinearity promotes uni-modal, uniformly distributed, periodic deformation patterns. In contrast, nonlinear effective softening behavior leads to subcritical post-bifurcation behavior, similarly divides the energy landscape by energy barriers and conversely promotes localization of deformations. With our theoretical model we can thus explain an experimentally observed phenomenon that in structures with effective softening followed by an effective stiffening behavior, the symmetry is initially broken by localizing the deformation and later restored by forming periodic, distributed deformation patterns as the load is increased. Finally, we show that initial imperfections can significantly alter the local or global energy-minimizing deformation pattern and completely remove some energy barriers. We envision that this knowledge can be extrapolated and exploited to convexify extremely divergent energy landscapes of more sophisticated systems, such as wrinkling compressed films on curved substrates (e.g., on cylinders and spheres) and that it will enable elementary analysis and the development of specialized numerical tools.