Implicit Formulation Research Articles

Influence of External Magnetic Field on 3D Thermocapillary Convective Flow in Various Thin Annular Pools Filled with Silicon Melt

Thermocapillary convection flows can have an impact on the homogeneity of floating zone semiconductor crystals. An external magnetic field can also help to reduce this non-homogeneity. The goal of this research is to minimize thermocapillary convection in various thin annular pools filled with silicon melt. A three-dimensional (3D) numerical technique is proposed that employs an implicit finite volume formulation. The steady-state thermocapillary flow in six thin annular pools (R=0.3, 0.4, 0.5, 0.6, 0.7, and 0.8) subjected to an externally induced magnetic field was observed. Under magnetic field influence, the effects of increasing annular gap, R on the hydrothermal wave number and azimuthal pattern are obtained. The results reveal that hydrothermal waves m=14, m=11, m=8, m=6, m=4, and m=3 are observed in steady flow for R=0.3; 0.4; 0.5; 0.6; 0.7, and R=0.8, respectively. The maximum temperature occurs in the intermediate zone between the inner and outer walls when there is no magnetic field. Under a strong enough magnetic field, isothermal lines change form and become concentric circles. As the amplitude of the magnetic field (Ha) grows, the azimuthal velocity and temperature at the free surface reduce, and the asymmetric 3D flow becomes axisymmetric steady when Ha surpasses a threshold value.

Simulation of hydrogen embrittlement of steel using mixed nonlocal finite elements

A new simulation strategy to model hydrogen embrittlement based on a multi-field finite element using displacements, pressure, volume variation, nonlocal damage variables and lattice hydrogen concentration as unknowns is proposed. The material is described using a modified GTN model, which includes the description of hydrogen enhanced decohesion (HEDE). The finite element problem is solved using a fully implicit formulation. Numerical problems related to volumetric locking are solved using a mixed pressure/volume variation formulation. The use of the mixed formulation allows a straightforward evaluation of the pressure gradient, which drives hydrogen diffusion. Mesh size dependence is solved using an implicit gradient nonlocal formulation. Two variables are used to represent damage: the plastic volume variation and the accumulated plastic strain, controlling nucleation. The model is used to simulate an existing experimental database (Moro et al., 2010; Briottet et al., 2012) including tensile, fracture toughness and pressurized disk tests. The model, after adjusting the various coefficients, can represent the main experimental findings: the effect of deformation rate on the failure of tensile specimens, the transition from surface to internal fracture with increasing deformation rate, the sharp toughness drop under hydrogen (CT specimen), the fracture location and the effect of the pressurization rate in the case of the tests on disks.

Coupling-strength criteria for sequential implicit formulations

The Fully Implicit Method (FIM) is used widely in the reservoir simulation community, especially for the so-called black-oil model. The primary reason for resorting to FIM is its unconditional stability, but this comes at the high computational cost of solving coupled nonlinear systems of all the conservation equations simultaneously. In addition, having a scalable FIM simulator relies on specialized multi-stage linear solvers. In the last few years, the Sequential Fully Implicit (SFI) formulation has received a great deal of attention. SFI decomposes the fully coupled nonlinear problem into two nonlinear sub-problems: flow and transport. The flow problem is solved implicitly for pressure, and the transport problem is solved implicitly for the saturation(s). The splitting usually assumes that the total-velocity is fixed during the transport computations. Because the pressure has a near-elliptic behavior and the saturation(s) have near-hyperbolic behavior, the flow-transport splitting of SFI allows for the use of dedicated scalable nonlinear and linear solvers for each sub-problem.At convergence, SFI and FIM lead to the same numerical solution within the specified tolerances on the residuals. SFI, however, can suffer from convergence difficulties of the overall nonlinear iterations between flow and transport. The convergence difficulties of SFI are intimately related to the splitting, and they depend on the type and strength of coupling between the nonlinear flow and transport problems. In this paper, we develop indicators that quantify the nature and strength of coupling across the systems of equations and solution variables. The indicators quantify the coupling in both space and time. Our strategy employs terms in the FIM Jacobian matrix to quantifying the sensitivity of the mass conservation equations and the global volume-balance to changes in pressure and saturation.These ‘coupling strength’ indicators are used in a domain-decomposition strategy to define local sub-domains for which a fully coupled correction is performed instead of the usual fixed-point iteration update. We demonstrate the accuracy and utility of these indicators for several challenging problems, including buoyancy driven lock-exchange - with or without gas solubility, as well as, gas and water injection problems in heterogeneous reservoir models.

High-order corrected trapezoidal rules for a class of singular integrals

We present a family of high-order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian nodes close to the singularity judiciously corrected based on the expansion. High-order accuracy can be achieved by utilizing a sufficient number of correction nodes around the singularity to approximate the terms in the series expansion. The derived quadratures are applied to the implicit boundary integral formulation of surface integrals involving the Laplace layer kernels.

Mixed Convection in a Square Cavity Filled with Porous Medium with Bottom Wall Periodic Boundary Condition

Transient mixed convection heat transfer in a confined porous medium heated at periodic sinusoidal heat flux is investigated numerically in the present paper. The Poisson-type pressure equation, resulted from the substituting of the momentum Darcy equation in the continuity equation, was discretized by using finite volume technique. The energy equation was solved by a fully implicit control volume-based finite difference formulation for the diffusion terms with the use of the quadratic upstream interpolation for convective kinetics scheme to discretize the convective terms and the temperature values at the control volume faces. The numerical study covers a range of the hydrostatic pressure sinusoidal amplitude range and time period values of . Numerical results show that the pressure contours lines are influenced by hydrostatic head variation and not affected with the sinusoidal amplitude and time period variation. It is found that the average Nusselt number decreases with time and pressure head increasing and decreases periodically with time and amplitude increasing. The time averaged Nusselt number decreases with imposed sinusoidal amplitude and cycle time period increasing.

An implicit and explicit dual model predictive control formulation for a steel recycling process

We present a formulation for both implicit and explicit dual model predictive control for a steel recycling process. The process consists in the production of new steel by choosing a combination of several different steel scraps with unknown pollutant content. The pollutant content can only be measured after a scrap combination is molten, allowing for inference on the pollutants in the different scrap heaps. The production cost should be minimized while ensuring high quality of the product through constraining the maximum amount of pollutant. The dual control formulation allows to achieve the optimal explore-exploit trade-off between uncertainty reduction and cost minimization for the examined problem. Specifically, the dual effect is obtained by considering the dependence of the future pollutant uncertainties on the scrap selection in the predictions. The implicit formulation promotes uncertainty reduction indirectly via the impact of active constraints on the objective, while the explicit formulation adds a heuristic cost on uncertainty to encourage active exploration. We compare the formulations by numerical simulations of a simplified but representative industrial steel recycling process. The results demonstrate the superiority of the two dual formulations with respect to a robustified but non-dual formulation.

#3379 DESIGN OF A NEW DIAPHRAGM PRESSURE TRANSDUCER FOR DIALYSIS MACHINE

Abstract Background and Aims Diaphragm pressure transducers can cause the dialysis process to be performed at higher quality. Pressure transduction is committed to concave, disc-shaped, impervious membranes that divide the respective chambers into two hermetically separate compartments: one contains liquid (blood/dialysate) while in the other there is trapped air. One of the sensitive parts in the design of a dialysis machine is diaphragm transducer which is used as a pressure gauge. One side of the diaphragm is pressurized, elastically deforming the diaphragm into a slightly curved shape and piezoelectric electrodes built a weak electrical signal that is used for measuring the pressure. The proper design of the diaphragm makes it sensitive to pressure changes and as a result, the pressure can be measured with high accuracy. Simulation of the flow inside the transducer can be used as a tool for designing the diaphragm and its housing. Method The behavior of whole blood is non-Newtonian. Under high shear forces, viscosity approaches an asymptotic value that corresponds to blood viscosity when considered as a Newtonian fluid. The Carreau-Yasuda model is used to express the relationship between shear stress and shear strain. Three modified and three novel geometries are suggested for pressure transducers. Each geometry is discretized with meshing software. Pyramidal cells are used to mesh the computing domain. Ansys Fluent software is an analyzer program that is used to simulate the field of fluid flow, heat transfer, etc. The solution is selected based on pressure. Due to the complexity of the geometry and convergence of the problem, only implicit formulation has been used for the model. The flow is assumed to be steady. The governing equation in this problem is continuity and momentum conservation and the k-ε model has been selected as for turbulent flow. Simple algorithm is considered for solving. Results The main output data is pressure distribution. Pressure and velocity distribution are related to each other. Figure 1 shows the velocity vector distribution in three modified schemes. The flow does not affect the pressure distribution inside the transducer housing for the straight port and is partially impressed in the shows-center port. The results of Figure 1 show that port location modification cannot affect the flow inside of the housing in such a way as to increase the sensitivity of the diaphragm. So, it was decided to redesign the geometry of the transducer's housing so can to apply the pressure variation on the diaphragm more sensibly. A curved part is added to the lower face of the transducer to guide the flow toward the diaphragm and distributed on the entire surface of the diaphragm to reach a uniform pressure. To find the best port location three configuration is used; i.e. straight, 90-degree, and 120-degree. Figure 2 is the velocity vector distribution for the other three novel geometries. In these schemes, flow circulates in the housing and has an effect directly on the diaphragm. The 90-degree port of the newly designed transducer has not had proper performance and pressure is concentrated on one side of the diaphragm which is between the ports. The uniformity of pressure becomes higher in the 120-degree. But in the straight scheme, the concentration is located exactly in the middle of the diaphragm and it is a desirable and optimal condition that the designer is seeking. Conclusion A series of simulations are carried out in the Ansys Fluent software to investigate a diaphragm pressure transducer flow after port locations and housing geometry modifications. The results show that if the fluid is directed to the diaphragm tangentially, the concentration of pressure is located on the diaphragm while the uniformity is reserved. The results show that this consideration leads to an optimal design using a new housing geometry with a curved shape wall to deflect the flow and distributed the pressure uniformly.

Simulating the entire rainfall-induced landslide process using the material point method for unsaturated soil with implicit and explicit formulations

Rainfall-induced landslides cause various types of damage, including damage to infrastructure, leading to devastating economic losses and human casualties. Although various numerical methods have been developed to predict landslide occurrence and the extent of sediment flow, three-dimensional analysis of the entire landslide process in a unified manner is still challenging owing to limitations in computational efficiency and the representation of deformation and flow. In this study, we present a numerical method of rainfall-induced landslides using the coupled hydromechanical material point method (MPM) for unsaturated porous media based on implicit and explicit formulations, which enables the efficient analysis of all rainfall-induced landslide processes, including both quasi-static and dynamic processes. The developed method includes an implicit MPM based on a simplified formulation, which is first applied to the quasi-static analysis in the pre-failure stages in which rainwater infiltrates the ground. The explicit MPM is then applied to the dynamic analysis for post-failure stages in which the ground collapses and flows. A constitutive law for soils is improved in the simulation of landslide initiation and sediment flow by incorporating the effect of cohesion in a visco-plastic model for granular materials. The proposed method was applied to a three-dimensional terrain model of Ashikita town, Kumamoto, Japan, where an actual landslide occurred owing to intense rainfall on July 3–4, 2020. Numerical results from the proposed hydromechanical coupling compared with single-phase MPM indicated that pore water plays an important role in understanding all rainfall-induced landslide processes, from landslide initiation to sediment discharge.

INFLUENCE OF THERMOPHYSICAL PROPERTIES ON TRANSFER OF HEAT IN MULTI-LAYER DUCTS AND TEMPERATURE CONTROL USING ELECTRIC HEATING

Ensuring oil production flow in offshore systems is a critical aspect of oil exploration operations. Any interruption in the production process, whether partial or complete, can result in significant financial losses and cause solid deposition in the production line. Such deposition is due to the crystallization of paraffin and hydrates, a common problem caused by low temperatures in deep waters. Among various mitigation strategies, the Pipe-in-Pipe (PIP) system with active heating is a technological solution to address this issue. This work aimed to perform a numerical simulation of the PIP system using the Finite Volume Method with an implicit formulation, considering the effect of temperature on fluid properties and the system's dynamic response. A control loop using a PI velocity algorithm was developed to maintain the temperature above the critical point. Such simulation studies were performed using the Python programming language in the Anaconda suite. The results showed that the fluid properties greatly influence the dynamic response. The PI control maintained the temperature in the desired condition, demonstrating its operational effectiveness in preventing solid deposition and delivering stable and low-oscillatory behavior. This research emphasizes the significance of taking temperature's impact on fluid properties into account when simulating offshore oil production systems and demonstrates the effectiveness of implementing feedback control.

MHD Effect on the Thermocapillary Silicon Melt Flow in Various Annular Enclosures

AbstractTo simulate thermocapillary magnetohydrodynamic (MHD) convection in different shallow annular enclosures filled with silicon melt, a 3D numerical approach using an implicit finite volume formulation is used. Radiation is emitted upward from the annular enclosure's free top surface, the bottom one heated vertically and is subjected to an external magnetic field. The steady and unsteady thermocapillary MHD flow in four annular enclosures (R = 0.5, 0.6, 0.7, and 0.8) field is studied. The effects of varying the annular gaps, R, on the hydrothermal wave number and azimuthal pattern are obtained. The effects of the magnetic field on azimuthal velocity, temperature disturbances, and the transition from oscillatory to steady flow are also depicted. The results reveal that: hydrothermal waves m = 4, m = 6 and m = 8 are observed in steady flow for R = 0.7, R = 0.6 and R = 0.5, respectively. Oscillatory flows dominated by a hydrothermal wave m = 3 are found also when R = 0.8. The azimuthal velocity and temperature fluctuations at the free surface decrease as the magnitude of the magnetic field (Ha) increases, and the oscillatory flow would turn steady when Ha exceeds a critical value.

Progressive Failure Analysis of Soil Slope with Strain Softening Behavior Based on Peridynamics

In this study, the nonlocal, nonordinary state-based peridynamics (NOSBPD) is introduced as a novel regularization technique to study the strain localization and progressive failure of soil slopes due to strain softening behavior. The NOSBPD formulations are introduced first, and then the strain-softening Mohr–Coulomb model is incorporated into the nonlocal framework. The implicit formulations of the nonlinear NOSBPD model under the plane strain condition combined with the energy dissipation-based arc-length method to avoid the limitations of the force and displacement control in the nonlinear analysis are given. Numerical examples, including the plane strain biaxial experiments and slope models, validate the effectiveness of the proposed method in terms of alleviating mesh sensitivity. The results show that this nonlocal method can remove the mesh dependence, and the localized shear band is related to the nonlocal parameter δ. The proposed method involves being able to capture strength mobilization in different parts of the sliding surface during the progressive failure process of the soil slope. It can avoid the dilemma that the peak strength is too dangerous and the residual strength is too conservative due to the rigid body assumption in the limit equilibrium method.

A New Analysis of an Implicit Mimetic Scheme for the Heat Equation

A mimetic finite difference scheme for the transient heat equation under Robin’s conditions is presented. The scheme uses second order gradient and divergence mimetic operators, on a staggered grid, to approximate the space derivatives. The temporal derivative is replaced by a first order backward difference approximation to obtain an implicit formulation. The resulting scheme contains nonstandard finite difference stencils. An original convergence analysis by the matrix’s method shows that the proposed scheme is unconditionally stable. A comparative study against standard finite difference schemes, based on central difference or first order one side approximations, reveals the advantages of our scheme without being its implementation more expensive or difficult to achieve.

Optimality equivalence and computational advantages of free-floating base dynamics compared to full-body dynamics

Several transcription methods exist for predictive simulations of free-floating base motions like athlete performing twisting somersaults. Following a brief overview of the literature, we proposed two original direct multiple shooting optimal control problems: (1) with a reduced set of equations motion (free-floating base dynamics) and generalized accelerations as controls; and (2) with the dynamics enforced implicitly through algebraic constraints and generalized jerks as controls. To assess their performance (computational time, dynamic consistency, cost), we compared full-body vs free-floating base dynamics and explicit vs implicit formulations of an optimal control problem of a double straight somersault with three twists. We found that free-floating dynamics is relevant when bounds on joint torques are not required since it hastens the convergence up to 10 times and the optimal solutions are similar. Implicit dynamics also speeds up optimization compared to the explicit formulation, but dynamic consistency may be affected. Using joint jerks instead of accelerations as controls mitigates this limitation, especially when the mesh grid densifies.

ANALYSIS OF MESOSTRUCTURE AND FRACTURE KINETICS OF ELEMENTS OF LATTICE COMPOSITE STRUCTURES UNDER TRANSVERSAL COMPRESSION USING STOCHASTIC FEA MICROMECHANICS

The paper analyses the mesostructure of the structural elements of lattice aircraft shells – ribs consisting of alternating layers of equal thickness and made from unidi-rectional CFRP and pure matrix material. In experimental studies, the elastic characteristics of unidirectional CFRP were obtained under three-point bending and transversal compression. As a result, the longitudinal modulus of elasticity of the layered composite turned out to be 101 GPa, and the shear modulus was 2.95 GPa. Numerical modeling of the meso- and micromechanics of the interaction of the noticed layers under transversal compression has been performed up to failure. The ANSYS FEA software (explicit and implicit formulations) was used. The regular and stochastic stacking of fibres in the cross section under compression is considered. The fiber diameters in the composite element were measured on thin sections using a Zeiss Axio Observer D1m digital microscope and were equal to 5.1 ± 0.8 µm. Layers with a fiber volume fraction of about 60 % alternate with layers of pure epoxy. It is proposed to use only the first principal stress in the matrix as a micromechanical criterion for failure under com-pression and tension. At the first stage of calculations, the problem of transversal compression of a cell with a regular laying of fibres was solved (the error in the value of the transversal modulus of elasticity was less than 2 %). At the second stage, an assessment was made of the strength and accumulation of microdamages under compression in a model of a layered structure with stochastic fibre stacking. The analysis of stress-strain state of a layered mesostructure under compression made it possible to explain the reason that the rib has a trans-verse strength twice lower than that of a homogeneous CFRP. The calculated values of the ultimate strength in transversal compression of a layered rib are in good agreement with the experimental ones.

Application of multicomponent sorption for diffusive transport in numerical simulation of tight reservoirs

One of the main challenges in modeling compositional transport in tight formations is to include the complex interaction of chemical species with rock surfaces. Such interactions result in compositional variations near the surfaces, which can significantly affect the compositional diffusive transport through tight pores.This paper presents the application of the Multicomponent Potential Theory of Adsorption (MPTA) for the fluid compositions in the central and near-wall sorbed layers subject to a wall chemical potential. The two-layer approximation of MPTA is implemented in the diffusive transport simulation that computes the mass transfer driven by fugacity gradient based on the dusty gas model for the central and sorbed layers of pores. This simulator uses a partially implicit formulation to solve the multiphase multicomponent mass transfer equations including the sorbed layer.The main novelty of this paper lies in the detailed analysis of the interplay between fluid components and rock surfaces, which became possible with the two-layer MPTA coupled with multicomponent diffusion. The diffusion simulation for a ternary case of methane, n-butane, and n-decane with the approximate MPTA shows that the sorption and capillary pressure can cause the compositional segregation between the central and sorbed layers. The segregation enhances the rates of methane injection and n-decane production in their counter-current diffusion. Methane (the lightest) is transported deep into the reservoir by diffusing through the central layer while n-decane (the heaviest) is diffused primarily through the sorbed layer. n-Butane (the intermediate) does not show preferential partitioning into either layer, resulting in relatively inefficient transport. In the absence of sorption and capillary pressure, the countercurrent diffusion occurs between methane (the lightest) and n-butane (the intermediate) while n-decane (the heaviest) remains nearly immobile. That is, whether the simulation considers the surface-fluid interactions can substantially affect the compositional transport (e.g., produced fluid composition) through tight porous media.The multicomponent Langmuir model is often used for computing the sorbed excess in micropores. However, results show that the Langmuir model becomes physically inconsistent at high pressures because it predicts a monotonically increasing function of pressure. The new approach developed in this research is sufficiently flexible and reasonably accurate for modeling multicomponent sorption at high pressures.

One-dimensional study of boundary effects and damage diffusion for regularized damage models

Local Continuum Damage Mechanics models cannot represent the entire degradation process in materials exhibiting strain-softening behaviors. It is well known that the rate equilibrium problem becomes ill-posed when softening occurs, and an infinity of solutions exists. From a numerical point of view, finite element analyses suffer from mesh-dependent results. Non-local models are generally used to regularize the structural response and recover objectivity. However, some physical inconsistencies can be observed in numerical results, e.g., damage diffusion over large damaged bands and damage attraction on the boundary of the computational domain. Non-local formulations with evolving interactions may better describe the damaging process and overcome these issues. This paper uses the so-called spalling test to underline the main drawbacks and advantages of several regularized models with constant and evolving non-local interactions. Concerning non-local formulations with constant interactions, attention is focused on the integral non-local formulation on the internal variable of the constitutive model and an implicit gradient damage formulation. Regarding formulations with evolving non-local interactions, attention is focused on a “stress-based integral non-local” approach and the so-called “eikonal non-local” approach. In this latter case, both its integral-type and gradient-type variants are considered.

Thermophysical improvement of PCM with inclusion of nanoparticles during solidification

Demonstration of heat release phenomenon by employing the numerical approach is the main purpose of current research. Water as PCM was combined by particles and homogeneous mixture was assumed. Various shapes of powder with different concentrations were employed. The unsteady energy equation involving nanomaterial properties and freezing source term has been analyzed and for finding the solution, the Galerkin technique was employed. The adaptive grid generates greater number of elements in solid front region. Implicit formulations for unsteady terms were implemented and automatic time step was employed in software. Solid front changes with alteration of shapes of nanopowder and its fraction. With fraction augmentation, freezing finishes in lower time. The needed time diminishes by about 10.29% and 13.78%, respectively. Changing the shape of particles to the biggest level makes the period decline by less than 4.8% and 8.4%. A greater fraction of nanomaterial leads to a higher effect on the shape of nanomaterial.

An implicit formulation of a two-grain cluster type homogenization approach for polycrystals

The crystal-plasticity-based finite element method has received considerable attention in the field of sheet-metal forming. When the size of stamped sheets is hundreds to thousands of millimeters, a homogenization approach with a lower computational cost is required. A previous study proposed a two-grain cluster (TGC)-type homogenization method, which allows relaxation from the fully-constrained Taylor model (Yoshida, 2022. An alternative formulation of a two-grain cluster model for homogenization of elastoviscoplastic behavior of polycrystal, International Journal of Plasticity, 156, 103368). In the present study, a fully implicit numerical procedure was developed for the TGC model, and it was applied to a cup drawing simulation. The present implicit TGC model used the fully implicit stress integration method. In addition, the relaxation vector that satisfies the equilibrium and compatibility conditions at the end of increment was determined. Moreover, the algorithmic tangent moduli that are consistent with the Newton–Raphson method were derived analytically. These features facilitate the implementation of the TGC model in the FEM. The developed model was implemented in static implicit FEM software (MSC Marc), and cylindrical cup drawing was simulated. Consequently, the FE simulation in conjunction with the TGC model was found to more accurately predict the cup height and thickness than that in conjunction with the Taylor model. The better prediction accuracy of R-values obtained by the TGC model contributed to the accurate prediction of the drawn cup profile.

An enhanced explicit–implicit time-marching formulation based on fully-adaptive time-integration parameters

In this paper, a new explicit–implicit time-marching formulation, which adapts to the properties of the model and to its computed responses, is proposed. In this novel mixed time-domain solution procedure, a group of three simple single-step recurrence relationships, which are formulated considering three time-integration parameters, is considered. These three time-integration parameters are here locally and adaptively computed and, as so, they are established for each element of the adopted spatial discretization procedure, considering optimized expressions that enable both enhanced numerically dissipative and non-dissipative analyses. Additionally, these self-adjustable parameters also define, taking into account entirely automated calculations, the explicit and implicit subdomains of the model, providing an explicit–implicit solution approach that is very easy to implement and to apply. At the end of the paper, numerical results are presented and compared to those of standard procedures, illustrating the excellent performance and effectiveness of the proposed new technique, which straightforwardly allows combining the best features (e.g., reduced solver efforts, guaranteed stability etc.) of both explicit and implicit formulations.

An implicit function formulation for optimization of discretized index-1 differential algebraic systems

A formulation for the optimization of index-1 differential algebraic equation systems (DAEs) is described that uses implicit functions to remove algebraic variables and equations from the optimization problem. The formulation uses the implicit function theorem to calculate derivatives of functions that remain in the optimization problem in terms of a reduced space of variables, allowing it to be solved with second-order nonlinear optimization algorithms. The formulation is shown to lead to more reliable solver convergence when compared with a full-space simultaneous formulation in challenging case studies involving a chemical looping combustion reactor. In a steady state simulation problem, the implicit function formulation solves 82 out of 100 instances, while the full space formulation solves only 42 out of 100 instances. In a dynamic optimization problem, the implicit function formulation solves 189 out of 200 instances, while the full space formulation solves only 152 out of 200 instances.