Selecting the simplest model for an analysis is not always trivial for engineers. A Hierarchic Modeling framework eases this burden by providing support for investigating model form errors.

In our previous SAFER Simulation articles, we have explored the concepts of Numerical Simulation, Challenges of Legacy FEA, Finite Element Modeling, Simulation Governance and High-Fidelity Aerostructure Analysis. We worked to establish a lexicon and foundational basis for how ESRD’s technological framework fits into solving the increasingly complex applications facing today’s engineering community.

In this SAFER Simulation article, we explore the concept of Hierarchic Modeling, some practical applications of Hierarchic Modeling, and the importance of implementing a Hierarchic Modeling framework in CAE software tools to support the practice of Simulation Governance.

What Is Hierarchic Modeling?

“Hierarchic models for plates and shells” by Drs. Ricardo Actis, Barna Szabó and Christoph Schwab. Comput. Methods Appl. Mech. Engrg. 172 (1999) 79-107.

The concept of Hierarchic Modeling is not new, it was introduced in the 1990s, and together with hierarchic finite element spaces and hierarchic basis functions it was implemented in StressCheck Professional. From the introduction to the 1999 Computer methods in applied mechanics and engineering technical paper “Hierarchic models of laminated plates and shells” by Drs. Actis, Szabó and Schwab:

The notion of hierarchic models differs from the notions of hierarchic finite element spaces and hierarchic basis functions. Hierarchic models provide means for systematic control of modeling errors whereas hierarchic finite element spaces provide means for controlling discretization errors. The basis functions employed to span hierarchic finite element spaces may or may not be hierarchic. Brief explanations follow:

Hierarchic models are a sequence of mathematical models, the exact solutions of which constitute a converging sequence of functions in the norm or norms appropriate for the formulation and the objectives of analysis. Of interest is the exact solution of the highest model, which is the limit of the converging sequence of solutions. In the case of elastic beams, plates and shells the highest model is the fully three-dimensional model of linear elasticity, although even the fully three-dimensional elastic model can be viewed as only the first in a sequence of hierarchic models that account for nonlinear effects, such as geometric, material and contact nonlinearities.

Hierarchic Modeling makes it possible to identify the simplest model that accounts for all features that influence the quantities of interest given the expected accuracy. This is related to the problem-solving principle, known as Occam’s razor, that when presented with competing models to solve a problem, one should select the model with the fewest assumptions, subject to the constraint of required accuracy.

Not all CAE software tools are capable of supporting Hierarchic Modeling in practice, especially for complex applications for which many modeling assumptions are to be examined.

What Do CAE Software Tools Need to Support Hierarchic Modeling?

As previously discussion in our S.A.F.E.R. Simulation blog article on Numerical Simulation, to enable support for a Hierarchic Modeling framework, and by extension the practice of Simulation Governance, CAE software tools must meet three basic requirements:

1. The model definition must be independent from the approximation.
2. Simple procedures must be available for assessing the influence of modeling assumptions (in support of model validation).
3. Simple procedures must be available for objective assessment of the errors of approximation (in support of solution verification).

The above requirements, and how meeting these requirements are supported in practice, are explained in greater detail in our Brief History of FEA page and its narrated video. The first implementation of model hierarchies in a CAE software tool, as explained in the video, was released in 1991 (ESRD’s StressCheck Professional).

An implementation framework meeting these three requirements enables the practice of Simulation Governance, providing the basis for the creation and deployment of engineering Sim Apps. Democratization of Simulation for standardization and automation of new technologies, such as Sim Apps, can be done with proper safeguards provided that the software tools used for the creation and deployment meet these technical requirements.

Why Should Engineers Care About Hierarchic Modeling?

“On the role of hierarchic spaces and models in verification and validation” by Drs. Barna Szabó and Ricardo Actis. Comput. Methods Appl. Mech. Engrg. 198 (2009) 1273-1280.

Legacy CAE tools used for Finite Element Modeling were not designed to support Hierarchic Modeling. This is because the concept of Hierarchic Modeling was established many years after the infrastructure of legacy FEA tools was created. Their main limitation is that the model definition and the approximation are not treated separately. Different orders of model complexity cannot be objectively compared by engineering analysts, therefore there is no basis for establishing  confidence in the modeling assumptions.

From 2009’s Computer methods in applied mechanics and engineering technical paper “On the role of hierarchic spaces and models in verification and validation” by Drs. Actis and Szabó:

…It is also necessary for the computer implementation
to support hierarchic sequences of models, allowing investigation of the sensitivities of the data of interest and the data measured in validation experiments to the various assumptions incorporated in the model…There is a strong predisposition in the engineering community to view each model class as a separate entity. It is much more useful however to view any mathematical model as a special case of a more comprehensive model, rather than a member of a conventionally defined model class.

For example, the usual beam, plate and shell models are special cases of a model based on the three-dimensional linear theory of elasticity, which in turn is a special case of large families of models based on the equations of continuum mechanics that account for a variety of hyperelastic, elastic-plastic and other material laws, large deformation, contact, etc. This is the hierarchic view of mathematical models.

Comparison of maximum von Mises stress convergence for different hierarchic fastened connection models.

To aid finding the simplest model, sensitivity studies via virtual experimentation are recommended. For example, modeling fastened joints may or may not require full multi-body contact effects if the data of interest are sufficiently far from the region of load transfer; bearing load applications, distributed normal springs or partial contact via “plugs” may be sufficient. By extension, if a structural support is to be approximated by distributed springs, the spring coefficients should be defined parametrically so that sensitivity studies are easy to perform.

The following examples and practical applications illustrate how a Hierarchic Modeling framework leads to increased control over and confidence in the engineering decision-making process.

Applications of Hierarchic Modeling In Engineering Practice

We will focus on two practical applications common to many aerospace engineers: fastened (bolted) joint analysis, and the influences of nonlinear effects such as plasticity. Both engineering applications require high-fidelity analyses to represent the data of interest, and are typically sensitive to the modeling assumptions.

Fastened Joint Analysis

ESRD recently provided a webinar titled “Hierarchic Approaches to Modeling Fastened Connections”, which incorporated the main points from the above discussion. The webinar can be viewed below in its entirety.

Through StressCheck Professional‘s Hierarchic Modeling framework, different modeling assumptions are tested for several classes of fastened joints and connections, including lap joints, splice joints and fittings, and in many cases a simpler model was found that represented the data of interest within a sufficient tolerance:

Some of the fastened joint modeling assumptions explored included the following:

• In-plane only vs out-of-plane bending effects on load transfer and detailed stresses
• 2D structural shear connections vs 3D detailed multi-body contact
• 2D bearing loading vs 3D bearing loading vs 3D multi-body contact
• Compression-only normal springs vs multi-body contact
• Fused fasteners vs multi-body contact
• Linear elastic vs. elastic-plastic materials

Without a Hierarchic Modeling framework, exploring these modeling assumptions would be unfeasible in engineering practice, and create a “simulation bottleneck” for engineering analysts.

Linear vs Nonlinear Effects

In some aerospace engineering applications, it may be necessary to investigate the influence of nonlinearities, such as plasticity and/or large deformations, in the results of interest.  For that reason a linear solution (i.e. small strain, small deformation and linear elastic material coefficients) must be viewed as the first in a hierarchy of models that includes nonlinear constitutive  relations, finite deformation and mechanical contact.

In a Hierarchic Modeling framework, engineering analysts should not need to change the discretization (i.e. mesh, element types and mapping) when transitioning from a linear to nonlinear model analysis for example; the switch should be seamless and simple, allowing the order of the model to increase on demand.

The following demo videos examine two case studies in nonlinear effects, in which the Hierarchic Modeling framework of StressCheck Professional was used to assess the influence of simplifying modeling assumptions without changing the discretization.

Geometric Nonlinearities

In the first case study, a linear vs. geometric nonlinear (large strain/large displacement) analysis for a 3D helical spring was performed:

Performing a geometric nonlinear analysis for the helical spring, in which equilibrium is satisfied in the deformed configuration, required no interaction with the model inputs or discretization parameters. The engineering analyst simply starts from a converged linear solution as the first step in the geometric nonlinear iterations.

The model hierarchies were then compared in live results processing with minimal effort, allowing the engineering analyst to quickly assess how accounting for large displacements/rotations affects the outcome of the results.

Material Nonlinearities

In the second case study, elastic-plastic materials are assigned to a detailed 3D eyebolt geometry, allowing plasticity to develop as the eyebolt is overloaded in tension:

To incorporate plasticity into the model, it was only required to update the material properties from linear to elastic-plastic; no other change in model inputs was required. Then, after a converged linear solution was available, a material nonlinear analysis was seamlessly initiated.

As in the previous case study, both models were available for assessment in live results processing, allowing the engineering analyst to determine whether material nonlinear effects are significant at the given load level (i.e. is the plasticity extensive) or the plastic zone is fully confined by elastic material.

Summary

As demonstrated in the above examples, and articulated in the technical paper excerpts, support for a seamless transition between model orders and theories is made possible by the implementation of a Hierarchic Modeling framework. To implement Hierarchic Modeling, CAE software tools must also allow separation of model definition from the discretization (in legacy FEA software, the definition of the model and the numerical approximation are combined, necessitating large element libraries). Without this clear separation, it is not feasible to reliably perform verification and validation in engineering practice.

Additionally, engineering analysts should expect modern FEA and CAE tools to support “what if” and sensitivity studies, such that modeling assumptions can be easily assessed and the simplest model used with confidence. As more and more engineering organizations look to democratize simulation, and virtual experimentation is increasingly used, it is essential to have numerical simulation tools that treat model definition separately from the approximation.

Finally, through the use of hierarchic finite element spaces and mathematical models it is possible to control approximation errors separately from modeling errors, while providing objective measures of solution quality for every result, anywhere in the model, in support of the increasing simulation demands on engineers.

How We Can Help…

ESRD’s Brent Lancaster discusses StressCheck at the ASIP 2018 conference.

This past week at ASIP 2018, ESRD provided a 2-hour training course titled “Modeling Fastened Connections: Hierarchic Approaches Discussion and Demo”, chatted with attendees about StressCheck and its ASIP-oriented strengths, and exhibited at our attractive booth (above image).

We enjoyed meeting many of the ASIP attendees, connecting with a wide variety of ESRD customers and colleagues, and continuing to share our perspective on DaDT and cold working analysis with the ASIP community. Thanks to those who stopped by to say hello (and grab one of our foam plane giveaways and/or handouts).

ESRD foam plane giveaway (it actually flies!)

ASIP 2018 Training Materials Available…

On November 26th, we had the pleasure of providing a training course on hierarchic modeling best practices (in the context A&D fastened joints) to a full house of interested ASIP engineers. Thanks to those who attended and/or requested the training course materials.

If you are interested in this fascinating topic (and investigating selected StressCheck models from our training course), you can download Mr. Lancaster’s ASIP training presentation, demo video and StressCheck model files here:

ASIP 2018 Training – Modeling Fastened Connections: Hierarchic Approaches Discussion and Demo

We are looking forward to receiving your feedback on our presentation and associated StressCheck model files.

Don’t have access to StressCheck? Request an evaluation of CAE Handbook, which will allow you to open and run the StressCheck model files.

Upcoming Webinar…

Are you looking for best practices and guidance for verifying the accuracy of detailed FEA results?  Don’t forget to sign up for our January 19th, 2019 webinar “How Do You Verify the Accuracy of Engineering Simulations?“:

In a previous edition of S.A.F.E.R. Simulation Views, we asked U.S. Navy NAVAIR Systems Command’s David Rusk about the challenges faced by A&D programs.

In this month’s S.A.F.E.R. Simulation Views we asked Brent Lancaster, Principal Support Engineer at ESRD, to discuss the basics of StressCheck Professional:

Q: What prompted the development of your software? What problem(s) is StressCheck meant to solve?

Brent:

Engineering Software Research and Development (ESRD) produces advanced engineering software, powered by higher-order, hierarchical finite element analysis solutions, and offers professional services in numerical simulation for the mechanical, aerospace and structural engineering industries.

Engineering analysis software should simultaneously improve the reliability of numerical solutions while reducing the time required for performing computer-aided engineering (CAE) tasks. This two-pronged strategy reduces costly physical testing and increases reliance on numerical simulation, saving the engineering community manufacturing, machining and design modification costs.

ESRD’s products and services fully embody this strategy, allowing us to establish the reliability of our engineering analyses with hassle-free solution verification and a hierarchic modeling framework that supports validation. ESRD’s flagship FEA software, StressCheck, wins in quality and efficiency by converging to the exact solution at a higher rate than our competitors, while simultaneously providing engineers crucial feedback that the model is fit for validation. Saving time and money while ensuring the quality of the answers, makes StressCheck the world’s premier high-definition, detailed numerical simulation tool.

StressCheck’s key solutions include detailed stress analysis, contact, global to local, laminated composites, fracture mechanics and engineered residual stresses due to cold-working, machining or other forming operations.

Q: What are the benefits of using StressCheck for fracture mechanics simulation?

Brent:

When performing fracture mechanics tasks, StressCheck users receive the high definition feedback information needed to establish the reliability and accuracy of the computed stress intensity factors (SIF), beta factors and energy release rates (ERR). This information is essential for accurate life prediction and damage tolerance assessment (DTA), a rapidly expanding need in the aerospace industry and beyond.

Some of the highlights of StressCheck’s fracture mechanics features are:

• Superconvergent SIF extractions – converges in less time that our competitors with fewer requirements for meshing.
• 3D crack insertion capability – model any crack shape, anywhere in a parametric or imported CAD part, with specialized meshing tools for accurate beta factors extraction.
• Point and click SIFs – get converged results anywhere along the 3D crack front to improve crack growth predictions.
• Combine global-local, contact and fracture mechanics seamlessly – no special elements, solvers or tricks required.
• Influence of residual stresses – incorporate coldworking, subsurface or bulk material residual stress effects when computing SIFs.

 

Q: Are there any unique applications that StressCheck works for that your competition cannot?

Brent:

Predicting fatigue-critical peak stresses and associated stress gradients in CAD parts with small and complex features is an application that is a huge challenge for our competitors. This is because their implementations are not designed to seamlessly quantify and separate discretization errors from modeling errors. Our competitors’ lack of this fundamental post-processing feedback limits the complexity of applications that can confidently be performed by their software, even for an application as basic as the one above.

As a result, many of ESRD’s customers rely on StressCheck as their go-to FEA solution for life extension, damage tolerance, plastic failure, fastened connections and laminated composites because they know they will get the correct answer.

Q: How much time does it take to learn and start using StressCheck?

Brent:

In order to establish the reliability of its solutions, StressCheck implements a more technologically rigorous approach to the finite element method (FEM) than our competitors. When coupled with a feature-rich interface supporting variable inputs and restriction-free outputs, StressCheck users will need more familiarity with the software in order to harness its full range of functionality.

If the application is as simple as importing a 3D solid, automeshing, applying material properties and BCs, solving a hierarchic series of increasingly refined solutions, plotting stress fringes and verifying the maximum stress converges to a limit, which is independent of the discretization, can be accomplished in less than an hour. In general, new StressCheck users can get up to speed in a manner of days or weeks, depending on the complexity of the application or the goals of the analysis.

See the above workflow performed in this step by step ESRD StressCheck Tutorial: Import CAD and Analyze Stress video.

More advanced or demanding applications, such as laminated composites, contact or cold-working analysis, may necessitate consulting the StressCheck Master Guide, visiting ESRD’s Customer Portal for best practices and demos, or hands-on training from a qualified ESRD representative.

To get started and become familiar with basic StressCheck features, we recommend first watching a broad-stroke overview of the StressCheck user interface and a sample workflow in this StressCheck Tutorial: GUI Walkthrough video.

Q: What are the biggest challenges or problems that customers in your target market face and how do you address their needs?

Brent:

Engineers involved in detailed analysis find that getting the right answer quickly and accurately is often difficult to achieve. Efficient simulations with error control are what make StressCheck stand out among the competition. StressCheck’s hierarchical modeling engine coupled with its separation of model definition (idealization) and numerical approximation helps ESRD’s clients find the correct answer quickly, an answer backed up automatically with data to show just how accurate within a range of tolerance (for example, an answer can be verified to be accurate within a margin of error of 2%).

Once the model is solved, engineers in many cases become interested in results different from what they originally had in mind.  In other words, they expect to have complete freedom in post-processing.  StressCheck allows users to live-query any location and any output within the area of interest, not requiring predefinition and eliminating the need to re-run the simulation if the user requires results outside the scope of the mesh design.

See an example of live-query results processing in this StressCheck Demo: Live Dynamic Extractions video.

Q: Describe a typical workflow using ESRD’s StressCheck.

Brent:

Imagine an engineer wants to quickly import a Parasolid assembly, incorporate a part-thru crack at a location, perform a multi-body contact analysis for more accurate interactions, and verify that the crack SIFs are converging well.  Here is the ideal workflow to complete the task:

• Import the Parasolid assembly using our file importation feature.
• Create or import the crack surface to be analyzed and insert into a flight-critical location in the assembly using the Boolean-Union method, resulting in a part-thru crack.
• Assign external loading (tension, bending, etc.) and constraints (fixed, spring, etc.) to the surfaces of the assembly and ensure that the assembly is properly constrained to prevent rigid body motion.
• Assign contact pairs to ensure the assembly transfers the load in contact regions.
• Tetmesh the assembly using curved elements and specialized meshing tools for the part-thru crack, including the Crack Face and Boundary Layer methods, for optimal grading.
• Define and assign material properties to the mesh.
• Solve the assembly via StressCheck’s automatic p-extension process, coupling its efficient contact algorithm with a powerful hierarchic element formulation, iterating until the difference between contact pressures of two consecutive iterations is sufficiently low.
• Check the quality of the load transfer and contact pressures to verify the solution is reliable.
• Plot the deformed shape to ensure that the part is deforming as expected, and the crack face is opening.
• Check convergence of SIFs at any location on the part-thru crack with fast Point and Click extractions.
• Query high-resolution gradients of SIF information along the crack front when multiple locations along the crack are used in crack growth.

 

See a similar workflow in action in this StressCheck Tutorial: 3D Elliptical Part-Thru Crack with Multi-Body Contact video.

Do you have a parametrized assembly, crack shape or loads?  Just change the relevant parameters and click Solve.  It’s really that simple: no tricks, shortcuts, or solver magic.

Q: What’s next for ESRD, what can we look forward to?

Brent:

There is an increasing demand in the engineering community for advanced capabilities in residual stress and crack extension analysis. Distortion due to machining and surface treatments such as shot peening are in need of refined solutions. With each StressCheck release, ESRD expands its analysis portfolio to account for challenges like these.

Speaking of StressCheck releases, we expect to release our next version, StressCheck v10.5, in Spring 2019. Be sure to subscribe to our newsletter to receive updates about our next product release.

For more information about ESRD’s StressCheck Professional, visit https://www.esrd.com/stresscheck-professional

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