A brand promise of durability (i.e. fitness for service over a suitable period) doesn’t mean much unless it is delivered reliably (i.e. with high consistency). When automakers provide a 100k mile warranty, for example, it is not enough to simply hit the promised life on average. Falling short of the promised life should occur only very rarely, if at all.
What effort can be justified in pursuing reliability? A quick way to estimate economic impact is to look at your product’s warranty adjustment rate. If your manufacturing contract is worth $10 million dollars / year, and your customer returns 1% of the product for premature failures, then you have an opportunity to save $100k / year by eliminating premature failures. This is a conservative estimate. If your early failure rate is notably higher than your competition’s, for example, you may find yourself losing contracts or being forced into price concessions that aren’t sustainable. A high failure rate may also result in legal liability for losses caused by your part. In this sense, the total value in achieving reliability can actually approach or even exceed the value of your business!
So in design, consider not only the expected life of the most common crack precursor for your material (half of the samples in your population will have shorter life than this!), but consider also the life of the rare oversized crack precursor that occurs 1 time in 100, or 1 time in 1 million. We recently launched a new Reliability Module to produce these statistics for exactly this purpose, check it out. Think of it as a way to put a probability-based “safety factor” on fatigue life predictions.
If you are involved in 3D printing with elastomers, can you predict the fatigue behavior?
How is product lifetime affected by complex lattice designs with multiaxial stresses, and what is the impact of printing defects?
Scientific literature and social media are abound with amazing examples of the potential for 3D printed articles made from metals, plastics and elastomers for use in many fields including the biomedical area. Researchers at ETH Zürich recently 3D printed a functioning artificial heart made from a silicone material. A picture of the device is shown below, and the story can be viewed elsewhere.1,2 This pioneering work represents a very noteworthy achievement. This research also highlights the importance of understanding elastomer durability in these cutting edge applications, as the silicone heart only survived 3,000 beats or about 30 minutes.
One of the key differences between 3D printing (additive manufacturing) and conventional manufacturing is the ability of 3D printing processes to create complex structures containing open spaces, often lattice-like in nature. Perhaps the most innovative and high profile example of a 3D printed product with lattice construction is the midsole for the Adidas Futurecraft 4D shoe that is created using the Carbon 3D technology.3
Overall stresses that are relatively modest and unidirectional translate into much higher stress, multiaxial conditions within the struts of a lattice structure like the shoe sole example above. The finite element simulation below illustrates this for a lattice structure undergoing simple compression (thanks to Mark Bauman, engineering analyst at Endurica).
Multiaxial load cases, crack closure considerations, and other complexities that arise in lattice designs and make it impossible to predict fatigue behavior using simplistic approaches such as Wohler / stress(S)-lifetime(N) curves, can be readily handled using the Endurica CL elastomer fatigue solver for Abaqus, MSC Marc, and ANSYS finite element analysis to predict when and where cracks will show up in the structure.
Cracks in an elastomer start out as microscopic precursors that grow due to applied cyclic loading according to a characteristic crack growth rate law for the material.4 In combination with critical plane analysis, this rubber fracture mechanics approach is the cornerstone of our Endurica CL software. The crack precursors – also called intrinsic defects or flaws – are especially important to pay attention to in the additive manufacturing of products in which voids or defects can be introduced by the printing process. The Core Module of our Fatigue Property Mapping testing services includes quantification of crack precursor size, and our new Reliability Module characterizes its distribution. The figure below illustrates the clear influence of crack precursor size on tensile strength in a study wherein we intentionally introduced glass microspheres as flaws in the rubber compound.5 Fatigue lifetime shows the same strong dependence on flaw size.
Endurica has the software, testing solutions, and expertise to help you understand and improve the durability of your 3D printed elastomer applications, so contact us to see how we can help you #GetDurabilityRight in the additive manufacturing world.
C. G. Robertson, L. B. Tunnicliffe, L. Maciag, M. A. Bauman, K. Miller, C. R. Herd, and W. V. Mars, “Characterizing Tensile Strength Distribution to Evaluate Filler Dispersion Effects and Reliability of Rubber”, paper presented at the Fall 196th Technical Meeting of the Rubber Division, American Chemical Society (International Elastomer Conference), Cleveland, OH, October 8-10, 2019.
Ever thought about what it takes to deliver the durability you expect from products you use? Durability reflects the combined sum of many decisions made all along the supply chain. What sources to use for raw materials? What dimensions and shape for product features? Are there OEM- or customer-imposed design constraints? What load cases occur in manufacturing, shipping, installation, and operation? Manufacturing processes? OEM-specified qualification and / or regulatory testing requirements? What is the warranty or brand promise? If these decisions are not made well, then durability (as well as cost and weight) will suffer.
The people making these decisions come from many backgrounds. They are chemists, product engineers, testing engineers, structural analysts. The big challenge is to organize things so that their contributions all add up to the desired end result: getting durability right, preferably on the first try. It’s a big challenge because the domain expertise and tools in place today in many organizations were largely built before the science was ready and before the workflows were understood well enough to integrate across disciplines. This situation can make it quite difficult to solve the durability puzzle. The pieces don’t all fit together!
Oversimplified lab tests whose relationship to actual product use is doubtful
Fatigue testing instruments that produce noisy data, or execute with uncontrolled test duration
Raw materials suppliers struggling to relate chemistry and process improvements to actual impact on end products
Compounders making materials selection decisions based on insufficient / poor information
Product engineers missing opportunities to fully leverage material capacity
Outdated and inaccurate ‘rule of thumb’ engineering that doesn’t work on new cases
Incomplete simulation efforts that fail to forecast or diagnose key durability issues
Product qualification tests that under- or over-solicit damage or change failure modes
Part suppliers leaving OEMs with too little confidence that durability issues have been handled
OEMs and part suppliers struggling to account for actual end-use load cases
Endurica-powered workflows overcome these barriers. Our training, testing services, testing instruments, and CAE software solutions integrate across disciplines. Our motto is “Get Durability Right”.
Our classes are geared specifically for your compounders, test engineers, product engineers and analysts. Your compounder doesn’t need to be a mechanical engineer, but she does need to negotiate the demands on the material. Your product engineer and your analyst don’t need a PhD in chemistry, but they do need to push for performance that will win for the customer. Your test engineer needs reliable, productive measurement strategies that get the key information that will power up your materials and product development efforts. Our classes will pay for themselves many times over when your team confronts the next durability pitfall.
Our testing services and testing instruments produce a complete picture of what limits durability in your application. Rubber exhibits many ‘special effects’, and our tests are very useful for quantifying each effect, for building material models, and for solving and diagnosing durability issues. We partner with leading labs around the world to bring you fast and reliable testing for durability simulation. We partner with testing instrument maker Coesfeld to bring our protocols directly to your own lab with automated, user-friendly control, measurement and data reduction. Analysts, designers and materials engineers all need clean, abundant, high-relevance measurements.
Our software (Endurica CL, Endurica DT, Endurica EIE and fe-safe/Rubber) provides the most complete set of durability analysis capabilities in the world. Total life, incremental damage, residual life, critical plane analysis, rainflow counting, nonlinear loads mapping, road load signal analysis, stiffness loss co-simulation, self-heating – its all here: documented, supported, validated, with examples and a large user-base. We support the Abaqus, Ansys and MSC/Marc Finite Element solvers. Use our software to see how different materials, different geometry, different load / use cases impact durability. If your materials, product, analysis or testing people can ask the question, chances are that our tools will simulate it and give you new insights.
Durability doesn’t have to be a difficult puzzle. It costs way too much when people from different disciplines don’t “speak the same language” and try to go forward with conflicting ideas and tools. Solve the puzzle by using pieces that fit together. Get your team speaking Endurican!
Keywords: Compounding, Design, Testing, Analysis, Training
Crack precursors exist in all elastomers owing to the heterogeneous microstructure, even before any loads are applied. The size of the typical precursor must be specified as part of the Endurica fatigue analysis workflow. The best practice for finding the precursor size is to leverage both crack growth and crack nucleation experiments to enforce agreement between the nucleation test results and the corresponding simulation-predicted life results. This procedure guarantees that both the crack growth and the crack nucleation experiments add up to an overall consistent story.
Prior to performing the calibration, you will need to have already defined the hyperelastic law, and the fatigue crack growth rate law. Fatigue models used for rubber have the following parameters:
Relationship between tearing energy and crack growth rate
The parameters needed to define this relationship are obtained through fatigue crack growth experiments. The crack is loaded under a range of tearing energies while tracking growth of the crack. These tests obtain the critical tearing energy, Tc, which is the tearing energy at which the crack reaches end of life failure in one loading. The crack growth rate at critical tearing energy, rc, and the slope of the curve, F, are determined by fitting a power law to the experimental crack growth and tearing energy.
This is the tearing energy limit T0 below which cracks do not grow. If you do not specify this parameter, then you will use the Thomas law. If you do specify this parameter, you will use the Lake-Lindley law. The threshold can be measured using an Intrinsic Strength experiment.
Some rubbers exhibit a strain crystallization behavior that causes an increase of durability under non-relaxing loads. If the duty cycle of your calibration experiment is nonrelaxing, and if you have a strain crystallizing material, then this characterization should be completed before calibrating the precursor size. The strain crystallization effect is measured in the non-relaxing module.
End of life crack size
This parameter should be set in the material definition prior to calibrating the precursor size. A default value of 1mm is generally adequate, particularly when it turns out that the precursor size is at least 5x smaller than this value. The part is considered to have failed when a crack reaches this size.
The crack nucleation experiment used for the calibration procedure may be made on a material test coupon, or on an actual component. Test coupons are convenient in early development stages as they do not require having a part to test. So long as crack precursor size is controlled by intrinsic features of the compound recipe (and not by the extrinsic features of post-mixing processes), a test coupon is likely to give useful results. There is a risk when using a test coupon: the risk that the precursor size in a manufactured part is actually controlled by some feature of post-mixing process such as factory contamination, part molding, abrasion, etc. This risk can be mitigated by calibrating precursor size on the basis of crack nucleation experiments on the finished part. In the following example, we show the process for calibration based on a finished part. The process for a test coupon is the same, but the model of the part is replaced by a model of the specimen.
To illustrate, take the case of a rubber bumper spring. Its duty cycle consists of compressing the 150 mm long rubber spring by 80 mm. Experiments show a fatigue life of 282,534 cycles for this duty cycle. A finite element analysis of the rubber spring is made to obtain strain history. The rubber spring is shown in the image below at the initial condition, at 50% of the displacement, and at 100% of the displacement during the fatigue duty cycle.
We are now ready to calibrate the as yet unknown precursor size to the known experimental fatigue test result of the spring. The precursor size can be calibrated by calculating the fatigue life for a series of precursor sizes and then interpolating to find the one precursor size that results in the best agreement between fatigue life calculations and the experimental fatigue life. Use the PRECURSORSIZE_CALIBRATION output request in Endurica CL to produce a table of fatigue life vs. crack precursor size. Your output request syntax will look something like this:
PRECURSORSIZE_CALIBRATION, NFS=25, FSMIN=1e-2 LIFE
NFS is the number of precursor sizes to evaluate, in this case 25. FSMIN is the smallest precursor size to evaluate, in this case 0.01 mm.
Once you’ve executed the calibration, use the new Endurica Viewer to complete the calibration workflow. It can plot a wide range of Endurica analysis outputs including precursor size calibration. Just open the Endurica output file containing the calibration results and expand the output file contents tree to find the Precursor Size Calibration results. The viewer then plots the computed table of precursor size vs fatigue life.
If you click on the plot options in the upper left corner, you can input the target life and the viewer will interpolate the precursor size. In this case, for a life of 282,534 cycles, the corresponding precursor size is 39 microns. Now that the precursor size is calibrated, the spring geometry can be optimized, different loadings analyzed, or entirely different parts can be analyzed using the material model to get fatigue life results that accurately reflect the precursor size that is most representative of the final material in the part. Again, if a part is not available, precursor size can also be calibrated to fatigue results from standard simple tension test specimen.
The calibrated rubber spring FE model with the life result of 282,534 cycles is shown below.
When people ask me what Endurica does I tell them: You give
us a computer file of one full use cycle of your design – be it a tire design
or one rotation of a pump that you’re building a seal for – along with a sample
of the rubber you’re making the product of and Endurica will tell you when it
will break and where. There are many companies who can do that for metals but
we’re the only ones who have figured it out for rubber. It all started with our
founder’s Ph.D. work in mechanical engineering and his years in tire design. We
actually have more clients outside of the U.S. than in, and our non-disclosure
agreements don’t allow us to share names but some of the clients who have
published technical papers using our software include General Motors, Caterpillar
I’ve learned that over-engineering seems to be the status
quo in the rubber industry. Because Endurica’s methods aren’t as well-known as
we would like, many companies do things the way they always have: test the
rubber part for a lifetime of use at the most intense conditions to ensure it
fails LONG past the time it could ever be used. That build-and-break routine is
so embedded in the industry it led to an interesting insight from an engineer
who stopped by our booth at a recent conference.
We don’t have time to do it right, but we do have time to do it over. – 2019 SAE World Congress Event Attendee
It seems the company they worked for budgets for five to
seven full development cycles (design, build, test to breaking point. Re-design;
build…..) I’m told that in the tire industry each round of this process easily tops $50,000 when you factor in the engineering
time, breaks in actual production schedules for samples to be made, plus months
in physical testing. It seems that because many do not understand Endurica’s
processes and the foundational science/engineering/technology behind it they
continue with the accepted norm of “make and break” even though it costs them
hundreds of thousands of dollars annually.
To prevent failure how much do YOU plan to fail?
If that is too strong of a question let me ask it this way:
How many design cycles do you have in the budget this year? Simulation is a
powerful tool in design and if you are designing on computer already, adding
Endurica’s methods to your simulations is the next logical step to, as we say,
Get Durability Right.
Consider using the same design budget you already have but
replace just one “round” of traditional design with the purchase of Endurica’s
training and a software license. By adding our software to your simulation
design system (Abaqus, ANSYS or MSC/Marc)
you can have results within HOURS (not the months of traditional
testing) for the durability of each version of your product design. Envision
the impact this technology could have on your firm: reduced time to market; greater
design flexibility, increased profitability; reduced costs in both engineering
If there was a better way, would you take it?
Endurica does not advocate that you go directly from simulation
to production. We simply make it easier for you to do MANY design cycles to get
the best design possible before you do actual FEA testing on the best possible
option. Maybe it’s time to reconsider your budget for design cycles, and factor
in budget money for both the training to thoroughly understand the science
behind Endurica’s methods as well as the software which will enable you to have
INFINITELY MORE design iterations for the same overall budget. It isn’t magic
but it is pretty advanced science and technology. Let’s talk.
Endurica CL received many improvements over the past year. These improvements cover a wide variety of
different aspects of the software:
Our investments in code benchmarking and performance are
paying off! We’ve been able to make internal optimizations to the code that
reduce analysis run-times by approximately 30%.
HFM and HFO Formatting
To make our output cleaner and more meaningful, small
changes have been made to the number formatting in the HFM and HFO files.
All results reported in scientific notation are now
formatted in standard form where the leading digit before the decimal point is
non-zero (previously the leading digit was always zero). This gives one more significant figure to all
the results without increasing the output file size.
The shortest fatigue life for the analysis is now printed to
the console and HFM file with six significant figures. Previously, the life was reported with only
two significant figures. This change
makes it easier to quickly compare two different analyses, especially when the
analyses have similar fatigue lives.
New features have been added to Endurica CL to make it
easier to process and analyze histories.
Using the new COMPRESS_HISTORY output request, you can generate new HFI
files containing compressed versions of your original history. The generated history is composed of the
rainflow counted cycles from your original history. An optional output parameter allows you to
further compress the signal by specifying the minimum percentage of the
original damage that should be retained in the new history. When keeping a percentage of the damage, the
cycles are sorted from most to least damaging so that the generated history
always contains the most damaging cycles and discards the least damaging
This output request is useful when you want to reduce a long
complex history while keeping the important damaging cycles. This can reduce file sizes and simplify
experimental testing setups as well as give you a deeper insight into your duty
Endurica DT is our incremental fatigue solver. With Endurica CL, your analysis starts at
time zero and integrates the given strain history until end-of-life. With Endurica DT, you can start and end at a
series of times that you specify. This
lets you accumulate many different histories and loading conditions repeatedly
Endurica DT gives you new ways to control your analyses, and
we have been using it over the past year in many applications. For example, fatigue results for laboratory
test procedures that involve multiple loading stages (such as FMVSS No. 139 for
light vehicle tires, or block cycle schedules for automotive component
applications) can be fully simulated using Endurica DT. You can also compute
residual life following some scheduled set of load cases.
Endurica DT can also be used to accumulate the actual loads measured
on a part in situ. This allows you to
create a digital twin that keeps a near real-time record of the part’s current
simulated damage state and the part’s remaining fatigue life.
Stiffness Loss Co-Simulation
Endurica DT now includes a stiffness loss co-simulation
workflow that allows you to iteratively update the stiffness of your part over
a series of time steps, based on the amount of damage occurring in the
part. The stiffness loss is computed per
element so you will have a gradient where the more damaged regions become
softer. Endurica DT computes the current
fraction h of stiffness loss based on the stress and strain, and the
finite element solver computes the stress and strain based on the current
fractions of stiffness loss. The capability accurately predicts the effects of
changing mode of control during a fatigue test.
For example, stress controlled fatigue tests show shorter life than
strain controlled fatigue tests.
Endurica EIE, our efficient interpolation engine, quickly
generates long, complex histories using a set of precomputed finite element
results (i.e. the ‘nonlinear map’). We
first launched EIE last year with the ability to interpolate 1-channel and
2-channel problems. We have recently
added the ability to interpolate 3-channel problems.
In the example below, EIE was benchmarked with
three-channels. Three separate road load
signals were computed from a single nonlinear map. With EIE, you don’t need to rerun the finite
element model for each history. Instead,
EIE interpolates from the nonlinear map, providing the equivalent results with
a 60x speed-up in compute time.
Ever wonder what it takes to consistently deliver quality and reliability in our software releases? Here’s a brief overview of the systems and disciplines we use to ensure that our users receive timely, trouble-free updates of Endurica software.
Throughout the life of our software, changes are made to our source code for a variety of reasons. Most commonly, we are adding new features and capabilities to our software. We also make updates to the code to improve performance and to squash the inevitable bugs that occasionally occur.
With each change committed to the code repository, the software needs to be built, tested, and released. Endurica’s workflow automates these steps so that any change to the source repository triggers a clean build of the software. A successful build is automatically followed by a testing phase where our suite of benchmarks is executed and compared to known results. Finally, the build is automatically packaged and stored so that it is ready to be delivered. At each step along the way, a build error or failed test will cancel the workflow and send an alert warning that the release has been rejected, so that the issue can be addressed, and the workflow restarted.
The automated testing phase that every release goes through helps ensure the reliability of our software. For example, every Endurica CL release must pass all 70 benchmarks. Each benchmark is a separate Endurica CL analysis made up of different materials, histories, and output requests. Results from a new build are compared to known results from the previous successful build. If results do not agree, or if there are any errors, the benchmark does not pass and the build is rejected.
The testing phase prevents “would-be” bugs from making it into a release and makes sure that any issues get resolved.
The automated nature of our development workflow naturally helps with repeatability in our releases. Each build flows through the same pipeline, creating consistent releases every time. There is less worry, for example, that a component will be forgotten to be included. It also allows us to recreate previous versions if comparisons need to be made.
Our version control system enables us to easily pinpoint where and when prior changes were introduced into the software. Each release is tied to a commit in the repository. This allows any future issues to be easily traced back and isolated to a small set of changes in the source for quick resolution.
Automating the build and release pipelines greatly increases our responsiveness. If an issue is discovered in a release, the problem can be resolved, and a fully corrected and tested release can be made available the same day. We can also quickly respond to user feedback and suggestions by making small and frequent updates.
The systems and disciplines we use in our development process make us very efficient, and they protect against many errors. This means we can spend more of our time on what matters: delivering and improving software that meets high standards and helps you to get durability right.
Q: With regards to fatigue life prediction methods, where was the rubber industry 10 years ago?
Will: There was plenty of great academic work and good understanding of fundamentals, but the methods were only deployed – if at all – via “homebuilt” solutions that could never support a broad enough audience to really impact daily product design decisions. Simulation methods and experimental methods shared theoretical foundations but they were poorly integrated. They suffered from operational problems, noisy data and open-ended test duration. It was possible to analyze a crack if you could mesh it, but the added bookkeeping and convergence burdens were usually not sustainable in a production engineering context. Mostly, analysts relied on tradition-based crack nucleation approaches that would look at quantities like strain or stress or strain energy density. These were not very accurate and they were limiting in many ways, even though they were widely used. They left companies very dependent on build and break iterations.
Q: Where is the industry today?
Will: The early adopters of our solutions have been off and running now for a number of years. Our critical plane method has gained recognition for its high accuracy when dealing with multiaxial cases, cases involving crack closure, cases involving strain crystallization. Our testing methods have gained recognition for high reliability and throughput. Our users are doing production engineering with our tools. They are consistently winning on durability issues. They are handling durability issues right up front when they bid for new business. They are expanding their in-house labs to increase testing capacity and they are winning innovation awards from OEMs. They are using actual road-load cases from their customers to design light-weight, just-right parts that meet durability requirements. The automotive industry has lead adoption but aerospace, tires, energy, and consumer products are also coming up. We have users across the entire supply chain: raw material suppliers, component producers and OEMs. The huge value that was locked up because durability was previously so difficult to manage is now unlocked in new ways for the first time. This has been the wind in Endurica’s sails for the last 10 years.
Q: Where do you see the industry in 10 years?
Will: In 10 years, OEMs will expect durability from all component producers on day 1, even for radical projects. They will expect designs already optimized for cost and weight. They will push more warrantee responsibility to the supplier. They will monitor durability requirements via shared testing and simulation workflows. Suppliers will pitch solutions using characterization and simulation to show their product working well in your product. The design and selection of rubber compounds to match applications will enter a golden age as real-world customer usage conditions will finally be taken fully into account. Where design and selection was previously limited by the budget for a few build and break iterations, and low visibility of design options, they will soon be informed by an almost unlimited evaluation of all possibilities. Where simulation methods have traditionally had greatest impact on product design functions, we will also start to see rubber part Digital Twins that track damage accumulation and create value in the operational functions of a business. Durability is definitely set to become a strong arena for competition in the next 10 years.
The accuracy of the interpolated results performed by EIE is dependent on the discretization of the map. Specifically, the results will become more accurate as the map’s point density increases. This study uses a simple 2D model to quantify the accuracy of results interpolated from maps with different densities.
A 1 mm x 1 mm rubber 2D plane strain model with two channels is used. The square’s bottom edge is fixed and the top edge is displaced in the x and y directions as shown below. The x displacement corresponds to channel 1 and the y displacement corresponds to channel 2. The working space of the model is defined by the x displacement ranging from 0 mm to 0.8 mm and the y displacement ranging from -0.08 mm to 0.8 mm.
The model is meshed with 100 8-node, quadrilateral, plane strain, hybrid, reduced integration elements (shown below).
We define as the benchmark reference solution a history that covers the model’s entire working space with a high density of points. An evenly spaced grid of 128×128 points for a total of 16384 points is used as the history (shown below). It is important that this history is more refined than the maps that we will create to ensure that we are testing all regions of our maps.
These points are used to drive the finite element model and the results are recorded. For this study, we record the three non-zero strain components and the hydrostatic pressure (NE11, NE22, NE12, and HP) for each element at each time point. In summary, there are 4 result components, 100 elements, and 16384 time increments. This set of results is the reference solution since it is solved directly by the finite element model. We will compare this solution to our interpolated results to measure our interpolation accuracy.
Six maps with different levels of refinement are used to compute interpolated results for our history points. All of the maps structure their points as an evenly spaced grid. The first map starts with two points along each edge. With each additional map, the number of points along each edge is doubled so that the sixth and final map has 64 edge points. The map points for the six maps are shown below.
The map points for these six maps are used to drive the finite element model’s two channels. The strain and hydrostatic pressure results from the FEA solutions are recorded at each map point in a similar way to how the results were recorded for the FEA solution that was driven by the history points. Next, EIE is used six times to interpolate the map point results at each resolution onto the high resolution reference history points.
We now have seven sets of history results: the true set of results and six interpolated sets of results.
To compare our results, we look at the absolute difference between the sets of results. The absolute error is used, opposed to a relative error, since some regions of the model’s working space will give near zero strain and hydrostatic pressure. Division by these near zero values would cause the relative error to spike in those regions.
Since we have 100 elements and 4 components per element, there are a lot of results that could be compared. To focus our investigation, we look at the element and component that gave the maximum error. The figure below shows contour plots for each of the six maps for this worst-case element and component. The component that gave the maximum error was NE12. The title of each of the contour plot also shows the maximum error found for each of the plots.
You can see that the error decreases as the map density increases. Also, you can identify the grid pattern in the contour plots since the error gets smaller near the map points.
Plotting the maximum error for each of the maps against the number of map points on a log scale is shown below. The slope of this line is approximately equal to 1 which is expected since a linear local interpolation was used to compute the results.
In considering ways to capture the contributions and essence of Endurica LLC to celebrate its tenth year of existence – and educating myself some more about the company I joined a little more than a year ago – I decided to put together the following top 10 list. Enjoy this informative snapshot of Endurica.
10 years of providing software and testing solutions for elastomer applications to #GetDurabilityRight in automotive, tire, aerospace, sealing, defense, consumer products, energy, and medical industries.