Deep learning based simulation of jack-up rig

The article illustrates the methodology of jack-up install simulation with the different compound of soils, including spudcan penetration. The core of this method is the combination of the finite element and data-driven approaches: the first - slow, but precise, and the second, based on machine learning proposed the fast solution with results, closing to the first approach.


Introduction
The jack-up vessels are widely used for the offshore works like: geological survey, oil and energy industries. Typical jack-up rig is a barge supported and lifted by three or four truss legs with a large spudcan on the end of each.
Jack-ups can be installed on a site independently, without external assistance. For the reaching of sufficient stability from the seabed in any conditions during installation, the ballast tanks are filled with seawater, this process is called preloading [1].
When the jack-up completed its work, the legs are retracted from the seabed, leaving a craters called 'footprints' at the site. Jack-ups often return to sites where previous operations have left footprint in the seabed [2].
In the process of jack-up vessel operations it faces a heavy environmental conditions: the axial loads due to the weight of the rig, the lateral and cyclic loads due to continuous waves and wind.
The most dangerous job is the jacking process, especially the sea bottom penetration part, that includes major risks, and therefore, should be analyzed and conducted, carefully [3].
The installation of a jack-up rig is a long time process, so the expand the jack-up operator's real experience much time is needed. The jack-up rig simulators provide much faster training for operators, and also can reproduce a wide range of events during jack-up works.
The bearing capacity of spudcan foundations for offshore mobile jack-up rigs is currently assessed using methods recommended by ISO [4] and SNAME [5], these recommendations for the punchthrough failure are the punching shear and projected area method. In the two methods, the peak resistance is calculated at a specified penetration depth by assuming an initial embedment of spud can and undisturbed soil strengths. The finite element method (FEM) has been introduced to the punchthrough failure analysis, especially the large deformation finite element (LDFE) method in recent years, such as the Remeshing and Interpolating Technique with Small Strain (RITSS), Arbitrary Lagrangian-Eulerian (ALE), and Coupled Eulerian-Lagrangian (CEL) [6]. The FEM are slow and need to setup the basic parameters of the simulation, these methods cannot be used for interactive simulations as needed for jack-up operator training, but the linear approximations of those can be used for interactive simulation.
The progress of data science allows for the application of data-driven approaches in the field of mathematical and computer modeling. Methods of big data, machine learning, neural networks, are used in science, engineering, chemistry, and pharmacy, etc. [7].
The application of such methods are well known in the simulation of transient processes [8], the simulation of turbulent and laminar flows of liquids and gases [9], in particular for solving the Navier-Stokes equations [10].
As a result of the application of data-driven methods neural networks that implements the necessary physical laws allow the fast and accurate simulation to be performed, which is suitable for training tasks. However, such methods have a natural flaw: the initial data are needed for work. A workaround for this disadvantage is the use of mixed methods, combining traditional modeling (numerical or analytical) and data-driven [11]. The result of this approach will be a fast simulation that is close in accuracy to traditional simulation.
In this article, we will look at a combination of FEM and machine learning for simulating jack-up rig installation and spudcan penetration.

Spudcan penetration finite element simulation
Spudcan foundations are generally under combined vertical (V), horizontal (H) and moment (M) loads due to the environmental loading (e.g., wind, waves, and currents). Figure 1 shows a platform during installation. The uncertainties and the complex load conditions make the bearing capacity estimation very difficult [12].  The Coupled Eulerian-Lagrangian (CEL) method from the Abaqus software was used to model the penetration of the spudcan into sea bottom soils. The Coupled Eulerian Lagrangian (CEL) method from the Abaqus software was used to model the penetration of the spudcan into sea bottom soils. To create the penetration model in Abaqus is needed to: make the spudcan and surrounding soils geometry, generate the mesh of finite elements, set the boundaries, the loads, and the displacements.
The simulation of the soil layers made from the different density clays is using the Mohr-Coulomb model of plastic fracture. Spudcans are modeled as solid materials, i.e. only Spudcan geometry and legs are involved in the analysis.
The "penalty contact method" including coefficient of friction is used for the algorithm of contact. There are important to set the model geometry, resolution of mesh, and speed of spudcan penetration.
An example of visualizing the result of FEM in the form of penetration depth from bearing capacity is presented in figure 3.

Subspace and data-driven simulation
The data-driven methods are based on precomputed data, that are typically accurate and computed not in real-time. The simulation of non-linear deformation without model reduction through a neural network is described in [13]. The data-driven methods are also used to simulate fluids [14][15]. Using spatially variable pressure data, a regression forest can predict the behavior of fluid particles [16][17]. This model predicts the dynamics of the individual particle in the mass of the particles, so the performance general requirement is the high level of parallelism, that achieves using GPU. The use of convolutional networks for selective acceleration allows solving incompressible Euler equations for fluid simulation [18].
The main problem of data-driven methods is keeping an optimal combination of performance, resource usage, accuracy, and model adequacy. The method, proposed in this article is close to the specified balance.
The method, illustrated in figure 4 includes four stages. The first stage is to collect accurate and quality FEM simulation data. The second stage is the use of machine learning with a neural network and a novel training method. After this result model is integrated into the jack-up rig simulator.
Principal component analysis (PCA) is one of the main ways to reduce the dimension of data while losing the least amount of information. From a mathematical point of view, this method is an orthogonal linear transformation that maps data from the original feature space to a new space of lower dimensions. A PCA basis is usable for the work with a representative set of deformation examples. An advantage of this method is the compatibility with almost any simulation method for acquiring the data, as the only input to our training procedure is a raw time-series of frame-by-frame vertex positions. In this example, data are taken from a FEM soft as the frames from animation, with the 10 6 frames for the best result.

Training
With the gathered data is possible to construct the training dataset, we collect the coordinates of the vertices in each frame into one vector ∈ ℝ 3 , and then we combine these frame-by-frame vectors into one large matrix = [ 0 , 1 , . . . , −1 ] ∈ ℝ 3 × , that describes the states of the modeled object. Also, the states of the external objects in the frame must be taken in count.
In result, there is one large vector for each object, representing the state of the external object ∈ ℝ , where is the number of degrees of freedom. All these vectors are also concatenated into the large matrix = [ 0 , 1 , . . . , −1 ] ∈ ℝ × .
After applying a PCA to and , the subspace representations and are constructed to generate subspace a image. = ( − ), = ( − ), where ∈ ℝ ×3 , ∈ ℝ × , u is the number of subspace bases, is the number of bases used to compress the external object representation, and is the mean values. If the PCA procedure requires too much memory, the data must be sampled first. Compression using PCA leads to loss of detail, but the high number (256) of subspace bases is usually enough to preserve most detail.
The development of the model that could predict the vectors states of the models in future frames: from −1 , −2 and . The simulated objects are usually characterized by inertia with a trend to some average rest state, and the model would be presented as: The and is the model parameters and ⊙ is a component-wise product. This parameters could be found by solving solving the least-squares equation individually for each dimension of and : where † denotes the matrix pseudo-inverse. ̅ is a very rough approximation of , without affect from external objects , and cannot be enough precise model for the training. So the neural network Φ should be trained to approximate extra effects of the model: (3) The neural network Φ is the standard feed-forward, with ten layers and ReLU [19] activation function (except the last layer because it is the output layer). For each layer between input and output is set the 1st International Conference on "Marine Geology and Engineering" IOP Conf. Series: Earth and Environmental Science 872 (2021) 012011 IOP Publishing doi:10.1088/1755-1315/872/1/012011 6 number of hidden units equals 1.5× size of PCA data, which gives a balance for the disk space and the performance.
Training a neural network through iterating over a dataset in order to obtain frame-wise predictions is not a good approach, due to unstable feedback. An alternative is the back propagation method used throughout the entire integration procedure (3), which provides stable long-term predictions.
The training procedure for one training sample is described below, where are network parameters, updatable by using the error, estimated for each step of object subspace state prediction: 0 , 1~( 0, ) (4) 0 * , 1 * ← 0 + 0 , 1 + 1 ℒ is the loss, and computed using mean absolute error: ℒ ← ℒ + ℒ (9) Updating network parameters using AmsGrad: ← ( , ∇ℒ) (10) The training procedure works like this: from the small sample of training data and took the first two frames 0 and 1 , then add a bit of noise 0 and 1 to offset the training trajectory. Then, to predict the next frames the algorithm (equation 6) is running several times, using previous prediction results at each new step. As the full trajectory is predicted the average coordinate and velocity errors are computed, and then passed to AmsGrad [20] optimizer to update the network parameters .
Using small training data sets forces the network to make corrections only within the sample window, thereby making the corrections small for all training data. This approach, presented in Figure 5, gives more stable corrections, avoiding overcorrection (typical for training on the whole data).
Repeating this algorithm on 16 frames from 32 frame sample, for 100 epochs or until training converges. The learning rate is 0.0001, learning rate attenuation of 0.999, and a standard noise deviation calculated from the first three components of PCA space. Training requires 48-96 hours, based on the complexity of the parameters and the PCA data size. On the left is the standard procedure, that gives an accurate result per frame, the predictions are unstable and give the overcorrection. On the right are proposed method, that targets an accurate prediction per frame, with the moderate corrections

Interactive application
The results of this method are visualized in our jack up rig simulator in the figure 6. This interactive application is written on C# with WPF, 3D render used DirectX. Pre-processing and neural network operations in single-threaded C++ code and load the binary network weights obtained during our training procedure.  Figure 6. Jack-up rig simulator

Discussion
The proposed mixed jack-up rig simulation method can be based not only on FEM but also on other modeling methods, as well as on pre-processed data from real rigs. Also, the resulting neural network can be adapted for calculating the jack-up loads and predicting its behavior in real-time during the installation process.

Conclusion
The mixture of two modeling approaches: a slow, but precise and the fast, but dependent on data is the novelty and the solution for domains, that used complex models and needs the fast results, like training simulations, predictions, etc.