The process of designing an appropriate drag force is given as an example here. Much of the code is indentical to existing constraints so would not actually be needed, it is included for completeness. Some things may have changed and also some boilerplate code is cut in the example so this may not work exactly in the form below. The user should refer to the examples in CPL_force.cpp and build up using this.

We will outline the process of designing the Ergun drag force,

`F = 150.0*e*nu*rho/(pow(e*D, 2.0)) + 1.75*rho()*Ur/(e*D)`

where we need to work out both porosity e and the fluid velocity Ur = U_MD-U_CFD from the particle simulation. We use an existing class as a basis, choose the one that is closest to your current case:

1) Let's choose a CPL_force type to inheret, the CPLForceDrag in this case. To define an inhereitance from, CPLForceGranular class, we use class CPLForceErgun : public CPLForceDrag and start by defining the class definintion in the header file CPL_force.h

```class CPLForceErgun : public CPLForceDrag {

public:

//Constructors
CPLForceErgun(CPL::ndArray<double>, map_strstr);
CPLForceErgun(int, int, int, int, map_strstr);

//Pre force collection and get force calculation
// position, velocity, acceleration, mass, radius, interaction
void pre_force(double r[], double v[], double a[],
double m, double s, double e);
std::vector<double> get_force(double r[], double v[], double a[],
double m, double s, double e);

//Flags for various input options (N.B. specify default values here)
bool calc_preforce = true;
bool use_overlap = false;
bool use_interpolate = false;
bool use_gradP = true;
bool use_divStress = false;
double mu = 0.0008900;
double rho = 1e3;

//All internal fields
std::shared_ptr<CPL::CPLField> nSums;
std::shared_ptr<CPL::CPLField> vSums;
std::shared_ptr<CPL::CPLField> eSums;
std::shared_ptr<CPL::CPLField> FSums;
std::shared_ptr<CPL::CPLField> FcoeffSums;

private:

double drag_coefficient();
void initialisesums(CPL::ndArray<double> f);
void resetsums();

};```

2) Now we have specified the header, we then need to override any routines which are unique to CPLForceErgun in the CPL_force.cpp file. But first, we need to define the constructors of our new class, these take in either the size of the grid array (local to a processes) or an existing array. These functions essentially creates the buffer which is used to store data from the CFD (in the parent). This is referred internally in functions of CPLForceErgun as "fieldptr" and the array data can be obtained. The child automatically calls the parent constructor and then we explicitly call initialisesums to setup various fields which will be populated pre_force. The logic here is the CPL array expected from the CFD code can be used and all other fields created with a consistent size.

```//Constructor using cells
CPLForceErgun::CPLForceErgun(int nd, int icells,
int jcells, int kcells,
map_strstr arg_map)
: CPLForceDrag(nd, icells, jcells, kcells, map_strstr arg_map)
{
//    unpack_arg_map(arg_map);
//    initialisesums(fieldptr->get_array());
}

//Constructor of datatype
CPLForceErgun::CPLForceErgun(CPL::ndArray<double> arrayin,
map_strstr arg_map)
: CPLForceDrag(arrayin, map_strstr arg_map){
//    unpack_arg_map(arg_map);
//    initialisesums(arrayin);
}```

Notice the input argmap with type map_strstr which is a map using strings defined in CPL_force.h.

typedef std::map <std::string, std::string> map_strstr

This is used to store all user keywords from the input line of LAMMPS (see above). We need to write the function to unpack these arguments so as to the important input values for your new CPL_force. Recall in the header we had a number of parameters:

```    //Flags for various input options (N.B. specify default values here)
bool use_overlap = false;
bool use_interpolate = false;
bool use_gradP = true;
bool use_divStress = false;
double mu = 0.0008900;
double rho = 1e3;```

We want to write unpack_arg_map to parse arg_map and over-ride default values with anything the user has specified. To do this, we can use a number of helper functions. We loop over arg_map using "for (const auto& arg : arg_map)" and can get keywords from arg.first and values from arg.second. We can then use string_contains(arg.first, keyword) to check if any part of the user input contains a keyword, for example "overlap", before setting the bool value in out CPL_force based on the value following the overlap keyword. We can use checktrue(arg.second) which handles 0/1 or case insensitive true/false returning the boolean value. This semi-manual method of setting insures the inputs specified are as required for the Force class we design. For numerical constants, we need to convert from string to double which is done using std::stod.

```    // Iterate over the map and print out all key/value pairs.
for (const auto& arg : arg_map)
{
if (string_contains(arg.first, "overlap") != -1) {
use_overlap = checktrue(arg.second);
} else if (string_contains(arg.first, "interpolate") != -1) {
use_interpolate = checktrue(arg.second);
} else if (string_contains(arg.first, "gradP")  != -1) {
} else if (string_contains(arg.first, "divStress")  != -1) {
use_divStress = checktrue(arg.second);
} else if (string_contains(arg.first, "mu")  != -1) {
mu = std::stod(arg.second);
} else if (string_contains(arg.first, "rho")  != -1) {
rho = std::stod(arg.second);
} else {
std::cout << "key: " << arg.first <<
" for forcetype not recognised " << '\n';
throw std::runtime_error("CPLForceDrag input not recognised");
}

}```

Note that calc_preforce is not set here. This determines if pre_force is called and this is not something the user should be able to change (it is essential to get eSum for the get_force calcultion and turning it off would cause problems). The initialisesums also needs to be developed, this can follow exactly from CPLForceDrag so we could simply inherit this, however it is here for completeness

```void CPLForceDrag::initialisesums(CPL::ndArray<double> arrayin){

int i = arrayin.shape(1);
int j = arrayin.shape(2);
int k = arrayin.shape(3);
nSums = std::make_shared<CPL::CPLField>(1, i, j, k, "nSums");
vSums = std::make_shared<CPL::CPLField>(3, i, j, k, "vSums");
eSums = std::make_shared<CPL::CPLField>(1, i, j, k, "eSums");
FSums = std::make_shared<CPL::CPLField>(3, i, j, k, "FSums");
FcoeffSums = std::make_shared<CPL::CPLField>(1, i, j, k, "FcoeffSums");

build_fields_list();
resetsums();
}

void CPLForceDrag::build_fields_list(){

fields.push_back(nSums);
fields.push_back(vSums);
fields.push_back(eSums);
fields.push_back(FSums);
fields.push_back(FcoeffSums);
}```

3) Next, we need to develop pre force and get_force for the Ergun equation. As Ergun needs porosity, this must be collected pre-force. The field classes provide an elegant way to get these (currently not used in most of cpl_force as this is new). We simply add to an array based on molecular position r, the value we want to, here a count for no. of molecules for nSums (1), the sphere volume for eSums and velocity vSums either weighted by overlap fraction in a cell or not. Again, nothing is needed here if the code below is sufficient, for Ergun this is the case.

```//Pre force collection of sums (should this come from LAMMPS fix chunk/atom bin/3d)
void CPLForceErgun::pre_force(double r[], double v[], double a[],
double m, double s, double e)
{
double vol = (4./3.)*M_PI*pow(s,3);
if (! use_overlap){
} else {
}
}```

If you wanted to use partial overlap calculation (fraction of sphere in box) then simply add the

```radius of the shere in as a second argument,
```

the choice between these is set by use_overlap from user input.

4) We now need to work out the value of force defined as a 3 vector. In principle, the only thing we need to change from CPLdrag is the drag coefficient routine here,

```double CPLForceErgun::drag_coefficient(double r[], double D) {
//Porosity e is cell volume - sum in volume
double e = 1.0 - eSums->get_array_value(r)/eSums.dV;
if (e < 1e-5) {
std::cout << "Warning: 0 particles in cell ("
<< cell << ", " << cell << ", " << cell << ")"
<< std::endl;
return 0.0;
}
return 150.0*e*nu*rho/(pow(e*D, 2.0)) + 1.75*rho/(e*D);
}```

Which is used as follows,

```//Get force using sums collected in pre force
std::vector<double> CPLForceErgun::get_force(double r[], double v[], double a[],
double m, double s, double e)
{
//Check array is the right size
CPL::ndArray<double>& array = cfd_array_field->get_array_pointer();
assert(array.shape(0) == 9);

//Get all elements of recieved field
if (! use_interpolate){
//Based on cell
std::vector<int> indices = {0,1,2};
Ui = cfd_array_field->get_array_value(indices, r);
for (int &n : indices) n += 3;
gradP = cfd_array_field->get_array_value(indices, r);
for (int &n : indices) n += 3;
divStress = cfd_array_field->get_array_value(indices, r);
} else {
//Or interpolate to position in space
std::vector<int> indices = {0,1,2};
Ui = cfd_array_field->get_array_value_interp(indices, r);
for (int &n : indices) n += 3;
gradP = cfd_array_field->get_array_value_interp(indices, r);
for (int &n : indices) n += 3;
divStress = cfd_array_field->get_array_value_interp(indices, r);
}

//Get Diameter
double D = 2.0*s;
//Get drag coefficient
double Cd = drag_coefficient(r, D);

//Calculate force
CPL::ndArray<int> indices = {0,1,2};
Ui = fieldptr->get_array_value(indices, r);
for (int i=0; i<3; i++){
f[i] = Cd*(Ui[i]-v[i]);
//Include pressure
// and stress
if (use_divStress)
f[i] += volume*divStress[i];
}

// Add sum of coefficients of forces
// Needed if you want to split implicit/explicit terms for
// improved numerical stability according to
// Xiao H., Sun J. (2011) Algorithms in a Robust Hybrid
// CFD-DEM Solver for Particle-Laden Flows,
// Commun. Comput. Phys. 9, 2, 297

return f;

}```

5) Finally, we need to add out new force type into the fix_cpl_force

```//Force factory
if (fxyzType.compare("Flekkoy") == 0) {
fxyz.reset(new CPLForceFlekkoy(9, cfdBuf->shape(1),
cfdBuf->shape(2),
cfdBuf->shape(3)));
...
...
...
} else if (fxyzType.compare("Ergun") == 0) {
fxyz.reset(new CPLForceErgun(9, cfdBuf->shape(1),
cfdBuf->shape(2),
cfdBuf->shape(3),
arg_map));```

and we can turn on by adding Ergun to the input line, something like this,

```   fix ID group-ID cpl/init region all forcetype Ergun overlap false interpolate true mu 0.0009 rho 1000 gradP true sendtype Granfull
```